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data-v-12048f0f><!--[--><p class="text" data-v-12048f0f><span data-v-12048f0f>密码学基础</span><!----></p><!--]--><!----></a><!----></div><!----></div><!--]--></div></div></section></div><div class="no-transition group" data-v-473fd05b><section class="vp-sidebar-item sidebar-item level-0 has-active" data-v-473fd05b data-v-12048f0f><div class="item" role="button" tabindex="0" data-v-12048f0f><div class="indicator" data-v-12048f0f></div><!----><h2 class="text" data-v-12048f0f><span data-v-12048f0f>古典加密算法</span><!----></h2><!----></div><div data-v-12048f0f data-v-12048f0f><div class="items" data-v-12048f0f><!--[--><div class="vp-sidebar-item sidebar-item level-1 is-link" data-v-12048f0f data-v-12048f0f><div class="item" data-v-12048f0f><div class="indicator" data-v-12048f0f></div><!----><a class="vp-link link link" href="/theory/cryptography/substitution-ciphers/" data-v-12048f0f><!--[--><p class="text" data-v-12048f0f><span data-v-12048f0f>替换密码</span><!----></p><!--]--><!----></a><!----></div><!----></div><div class="vp-sidebar-item sidebar-item level-1 is-link" data-v-12048f0f data-v-12048f0f><div class="item" data-v-12048f0f><div class="indicator" data-v-12048f0f></div><!----><a class="vp-link link link" href="/theory/cryptography/permutation-encryption/" data-v-12048f0f><!--[--><p class="text" data-v-12048f0f><span data-v-12048f0f>置换密码</span><!----></p><!--]--><!----></a><!----></div><!----></div><!--]--></div></div></section></div><!--]--><!--[--><!--]--></nav></aside><!--[--><div id="VPContent" vp-content class="vp-content has-sidebar" data-v-f73ca3da data-v-b2beaca7><div class="vp-doc-container has-sidebar has-aside" data-v-b2beaca7 data-v-23f6ad98><!--[--><!--]--><div class="container" data-v-23f6ad98><div class="aside" vp-outline data-v-23f6ad98><div class="aside-curtain" data-v-23f6ad98></div><div class="aside-container" data-v-23f6ad98><div class="aside-content" data-v-23f6ad98><div 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data-v-1ae4ad7a><a class="vp-link link breadcrumb" href="/" property="item" typeof="WebPage" data-v-1ae4ad7a><!--[-->首页<!--]--><!----></a><span class="vpi-chevron-right" data-v-1ae4ad7a></span><meta property="name" content="首页" data-v-1ae4ad7a><meta property="position" content="1" data-v-1ae4ad7a></li><li property="itemListElement" typeof="ListItem" data-v-1ae4ad7a><span class="vp-link breadcrumb" property="item" typeof="WebPage" data-v-1ae4ad7a><!--[-->古典加密算法<!--]--><!----></span><span class="vpi-chevron-right" data-v-1ae4ad7a></span><meta property="name" content="古典加密算法" data-v-1ae4ad7a><meta property="position" content="2" data-v-1ae4ad7a></li><li property="itemListElement" typeof="ListItem" data-v-1ae4ad7a><a class="vp-link link breadcrumb current" href="/theory/cryptography/substitution-ciphers/" property="item" typeof="WebPage" data-v-1ae4ad7a><!--[-->替换密码<!--]--><!----></a><!----><meta property="name" content="替换密码" data-v-1ae4ad7a><meta property="position" content="3" data-v-1ae4ad7a></li><!--]--></ol></nav><!--[--><!--]--><!--[--><h1 class="vp-doc-title page-title" data-v-ba8d1a1e><!----> 替换密码 <!----></h1><div class="vp-doc-meta" data-v-ba8d1a1e><!--[--><!--]--><p class="reading-time" data-v-ba8d1a1e><span class="vpi-books icon" data-v-ba8d1a1e></span><span data-v-ba8d1a1e>约 1648 字</span><span data-v-ba8d1a1e>大约 5 分钟</span></p><!----><!--[--><!--]--><p class="create-time" data-v-ba8d1a1e><span class="vpi-clock icon" data-v-ba8d1a1e></span><span data-v-ba8d1a1e>2025-10-27</span></p></div><!--]--><!--[--><!--]--><div class="_theory_cryptography_substitution-ciphers_ external-link-icon-enabled vp-doc plume-content" vp-content data-v-23f6ad98><!--[--><!--]--><div data-v-23f6ad98><p>我们一起来系统梳理古典加密算法（Classical Ciphers）。这些算法虽然在现代已不再安全，但它们是密码学发展的基石，蕴含了替换、置换、密钥等核心思想，非常适合理解密码学的基本原理。</p><p>替换密码的核心思想是“一对一”或“多对一”的字符映射：把明文中的每一个字母（或符号）按照事先约定好的规则，替换成另一个字母（或符号）。</p><p>这种映射可以是固定不变的（如凯撒密码的“统一移位”），也可以是依赖密钥动态变化的（如维吉尼亚密码的“周期移位”）。</p><p>由于密文保留了原始字母的出现频率，只是“换了一张皮”，所以替换密码在本质上没有改变字母的统计特性，这也为频率分析攻击留下了突破口。</p><p>替换操作可以手工完成，也可以通过查表、转盘、甚至机械电路实现，是后续更复杂多表替换与乘积密码的雏形。</p><h2 id="一、凯撒密码-caesar-cipher" tabindex="-1"><a class="header-anchor" href="#一、凯撒密码-caesar-cipher"><span>一、凯撒密码（Caesar Cipher）</span></a></h2><p><strong>工作原理</strong>： 凯撒密码是一种循环移位密码，将字母表视为一个环形结构。加密时每个字母向后移动固定位置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，解密时向前移动相同位置。</p><p><strong>数学表示</strong>： 设字母 A-Z 对应数字 0-25，则：</p><p>加密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn></mrow><annotation encoding="application/x-tex">E(x) = (x + k) \mod 26 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span></span></span></span></span></p><p>解密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn></mrow><annotation encoding="application/x-tex">D(x) = (x - k) \mod 26 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 是明文字母编号，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 是密钥（0 ≤ k ≤ 25）。</p><p><strong>特点</strong>：</p><ul><li>实现简单，易于理解</li><li>密钥空间仅 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>26</mn></mrow><annotation encoding="application/x-tex">26</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">26</span></span></span></span> 种可能，安全性极低</li><li>易受频率分析攻击</li><li>主要具有教学价值</li></ul><h2 id="二、单表替换密码-simple-substitution-cipher" tabindex="-1"><a class="header-anchor" href="#二、单表替换密码-simple-substitution-cipher"><span>二、单表替换密码（Simple Substitution Cipher）</span></a></h2><p><strong>工作原理</strong>： 单表替换密码是凯撒密码的泛化形式，它使用一个随机的字母替换表，而不是固定的移位。每个明文字母都被唯一地映射到一个密文字母，形成一对一的替换关系。</p><!--[--><div class="mermaid-actions"><button class="preview-button" title="preview"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1316 1024" fill="currentColor"><path d="M658.286 0C415.89 0 0 297.106 0 512c0 214.82 415.89 512 658.286 512 242.322 0 658.285-294.839 658.285-512S900.608 0 658.286 0zm0 877.714c-161.573 0-512-221.769-512-365.714 0-144.018 350.427-365.714 512-365.714 161.572 0 512 217.16 512 365.714s-350.428 365.714-512 365.714z"/><path d="M658.286 292.571a219.429 219.429 0 1 0 0 438.858 219.429 219.429 0 0 0 0-438.858zm0 292.572a73.143 73.143 0 1 1 0-146.286 73.143 73.143 0 0 1 0 146.286z"/></svg></button><button class="download-button" title="download"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024" fill="currentColor"><path d="M828.976 894.125H190.189c-70.55 0-127.754-57.185-127.754-127.753V606.674c0-17.634 14.31-31.933 31.933-31.933h63.889c17.634 0 31.932 14.299 31.932 31.933v95.822c0 35.282 28.596 63.877 63.877 63.877h511.033c35.281 0 63.877-28.595 63.877-63.877v-95.822c0-17.634 14.298-31.933 31.943-31.933h63.878c17.635 0 31.933 14.299 31.933 31.933v159.7c0 70.566-57.191 127.751-127.754 127.751zM249.939 267.51c12.921-12.92 33.885-12.92 46.807 0l148.97 148.972V94.893c0-17.634 14.302-31.947 31.934-31.947h63.876c17.638 0 31.946 14.313 31.946 31.947v321.589l148.97-148.972c12.922-12.92 33.876-12.92 46.797 0l46.814 46.818c12.922 12.922 12.922 33.874 0 46.807L552.261 624.93c-1.14 1.138-21.664 13.684-42.315 13.693-20.877.01-41.88-12.542-43.021-13.693L203.122 361.135c-12.923-12.934-12.923-33.885 0-46.807l46.817-46.818z"/></svg></button></div><div class="mermaid-wrapper"><div style="display:flex;align-items:center;justify-content:center;height:96px;" class="mermaid-loading"><span style="--loading-icon: url("data:image/svg+xml;utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' preserveAspectRatio='xMidYMid' viewBox='25 25 50 50'%3E%3CanimateTransform attributeName='transform' type='rotate' dur='2s' keyTimes='0;1' repeatCount='indefinite' values='0;360'%3E%3C/animateTransform%3E%3Ccircle cx='50' cy='50' r='20' fill='none' stroke='currentColor' stroke-width='4' stroke-linecap='round'%3E%3Canimate attributeName='stroke-dasharray' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='1,200;90,200;1,200'%3E%3C/animate%3E%3Canimate attributeName='stroke-dashoffset' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='0;-35px;-125px'%3E%3C/animate%3E%3C/circle%3E%3C/svg%3E");--icon-size: 48px;display: inline-block;width: var(--icon-size);height: var(--icon-size);background-color: currentcolor;-webkit-mask-image: var(--loading-icon);mask-image: var(--loading-icon)"></span></div></div><!--]--><p><strong>数学表示</strong>： 设字母表 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Σ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">,</mo><mi>C</mi><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mi>Z</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Sigma = \{A,B,C,...,Z\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">}</span></span></span></span>，替换函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi mathvariant="normal">Σ</mi><mo>→</mo><mi mathvariant="normal">Σ</mi></mrow><annotation encoding="application/x-tex">f: \Sigma \rightarrow \Sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Σ</span></span></span></span> 是一个双射（一一对应），则：</p><p>加密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(x) = f(x) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><p>解密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(y) = f^{-1}(y) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 的逆函数。</p><p><strong>密钥空间</strong>： 单表替换密码的密钥空间是所有可能的字母排列，大小为：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>K</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>26</mn><mo stretchy="false">!</mo><mo>≈</mo><mn>4.03</mn><mo>×</mo><msup><mn>10</mn><mn>26</mn></msup></mrow><annotation encoding="application/x-tex">|K| = 26! \approx 4.03 \times 10^{26} </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">26</span><span class="mclose">!</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4.03</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">26</span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>这个巨大的密钥空间使得暴力破解在计算上不可行。</p><p><strong>示例</strong>： 假设替换表为：</p><div class="language- line-numbers-mode" data-highlighter="shiki" data-ext="" style="--shiki-light:#393a34;--shiki-dark:#dbd7caee;--shiki-light-bg:#ffffff;--shiki-dark-bg:#121212;"><pre class="shiki shiki-themes vitesse-light vitesse-dark vp-code"><code class="language-"><span class="line"><span>A→Q, B→W, C→E, D→R, E→T, F→Y, G→U, H→I, I→O, J→P,</span></span>
|
||
<span class="line"><span>K→A, L→S, M→D, N→F, O→G, P→H, Q→J, R→K, S→L, T→Z,</span></span>
|
||
<span class="line"><span>U→X, V→C, W→V, X→B, Y→N, Z→M</span></span></code></pre><div class="line-numbers" aria-hidden="true" style="counter-reset:line-number 0;"><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div><!--[--><div class="mermaid-actions"><button class="preview-button" title="preview"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1316 1024" fill="currentColor"><path d="M658.286 0C415.89 0 0 297.106 0 512c0 214.82 415.89 512 658.286 512 242.322 0 658.285-294.839 658.285-512S900.608 0 658.286 0zm0 877.714c-161.573 0-512-221.769-512-365.714 0-144.018 350.427-365.714 512-365.714 161.572 0 512 217.16 512 365.714s-350.428 365.714-512 365.714z"/><path d="M658.286 292.571a219.429 219.429 0 1 0 0 438.858 219.429 219.429 0 0 0 0-438.858zm0 292.572a73.143 73.143 0 1 1 0-146.286 73.143 73.143 0 0 1 0 146.286z"/></svg></button><button class="download-button" title="download"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024" fill="currentColor"><path d="M828.976 894.125H190.189c-70.55 0-127.754-57.185-127.754-127.753V606.674c0-17.634 14.31-31.933 31.933-31.933h63.889c17.634 0 31.932 14.299 31.932 31.933v95.822c0 35.282 28.596 63.877 63.877 63.877h511.033c35.281 0 63.877-28.595 63.877-63.877v-95.822c0-17.634 14.298-31.933 31.943-31.933h63.878c17.635 0 31.933 14.299 31.933 31.933v159.7c0 70.566-57.191 127.751-127.754 127.751zM249.939 267.51c12.921-12.92 33.885-12.92 46.807 0l148.97 148.972V94.893c0-17.634 14.302-31.947 31.934-31.947h63.876c17.638 0 31.946 14.313 31.946 31.947v321.589l148.97-148.972c12.922-12.92 33.876-12.92 46.797 0l46.814 46.818c12.922 12.922 12.922 33.874 0 46.807L552.261 624.93c-1.14 1.138-21.664 13.684-42.315 13.693-20.877.01-41.88-12.542-43.021-13.693L203.122 361.135c-12.923-12.934-12.923-33.885 0-46.807l46.817-46.818z"/></svg></button></div><div class="mermaid-wrapper"><div style="display:flex;align-items:center;justify-content:center;height:96px;" class="mermaid-loading"><span style="--loading-icon: url("data:image/svg+xml;utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' preserveAspectRatio='xMidYMid' viewBox='25 25 50 50'%3E%3CanimateTransform attributeName='transform' type='rotate' dur='2s' keyTimes='0;1' repeatCount='indefinite' values='0;360'%3E%3C/animateTransform%3E%3Ccircle cx='50' cy='50' r='20' fill='none' stroke='currentColor' stroke-width='4' stroke-linecap='round'%3E%3Canimate attributeName='stroke-dasharray' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='1,200;90,200;1,200'%3E%3C/animate%3E%3Canimate attributeName='stroke-dashoffset' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='0;-35px;-125px'%3E%3C/animate%3E%3C/circle%3E%3C/svg%3E");--icon-size: 48px;display: inline-block;width: var(--icon-size);height: var(--icon-size);background-color: currentcolor;-webkit-mask-image: var(--loading-icon);mask-image: var(--loading-icon)"></span></div></div><!--]--><p><strong>安全性分析</strong>： 虽然单表替换密码的密钥空间巨大，但它仍然易受<strong>频率分析攻击</strong>。因为：</p><ol><li><strong>字母频率保留</strong>：高频字母（如E、T、A）在密文中仍然是高频</li><li><strong>单词模式保留</strong>：常见单词模式（如"THE"、"ING"）在密文中保持相同模式</li><li><strong>双字母频率</strong>：常见字母对（如"TH"、"ER"）的频率特征仍然存在</li></ol><p><strong>攻击方法</strong>：</p><ul><li>单字母频率分析</li><li>双字母频率分析</li><li>单词长度和模式分析</li><li>已知明文攻击</li></ul><p><strong>特点</strong>：</p><ul><li>密钥空间巨大（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>26</mn><mo stretchy="false">!</mo></mrow><annotation encoding="application/x-tex">26!</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord">26</span><span class="mclose">!</span></span></span></span>），理论上难以暴力破解</li><li>仍然易受统计攻击</li><li>是密码学历史上重要的里程碑</li><li>为现代密码学提供了重要启示</li></ul><h2 id="三、维吉尼亚密码-vigenere-cipher" tabindex="-1"><a class="header-anchor" href="#三、维吉尼亚密码-vigenere-cipher"><span>三、维吉尼亚密码（Vigenère Cipher）</span></a></h2><p><strong>工作原理</strong>： 维吉尼亚密码是一种多表替换密码，它使用一个关键词来决定每次替换的凯撒密码移位量。关键词的每个字母对应一个移位量，明文的每个字母根据关键词的循环使用进行替换。</p><!--[--><div class="mermaid-actions"><button class="preview-button" title="preview"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1316 1024" fill="currentColor"><path d="M658.286 0C415.89 0 0 297.106 0 512c0 214.82 415.89 512 658.286 512 242.322 0 658.285-294.839 658.285-512S900.608 0 658.286 0zm0 877.714c-161.573 0-512-221.769-512-365.714 0-144.018 350.427-365.714 512-365.714 161.572 0 512 217.16 512 365.714s-350.428 365.714-512 365.714z"/><path d="M658.286 292.571a219.429 219.429 0 1 0 0 438.858 219.429 219.429 0 0 0 0-438.858zm0 292.572a73.143 73.143 0 1 1 0-146.286 73.143 73.143 0 0 1 0 146.286z"/></svg></button><button class="download-button" title="download"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1024 1024" fill="currentColor"><path d="M828.976 894.125H190.189c-70.55 0-127.754-57.185-127.754-127.753V606.674c0-17.634 14.31-31.933 31.933-31.933h63.889c17.634 0 31.932 14.299 31.932 31.933v95.822c0 35.282 28.596 63.877 63.877 63.877h511.033c35.281 0 63.877-28.595 63.877-63.877v-95.822c0-17.634 14.298-31.933 31.943-31.933h63.878c17.635 0 31.933 14.299 31.933 31.933v159.7c0 70.566-57.191 127.751-127.754 127.751zM249.939 267.51c12.921-12.92 33.885-12.92 46.807 0l148.97 148.972V94.893c0-17.634 14.302-31.947 31.934-31.947h63.876c17.638 0 31.946 14.313 31.946 31.947v321.589l148.97-148.972c12.922-12.92 33.876-12.92 46.797 0l46.814 46.818c12.922 12.922 12.922 33.874 0 46.807L552.261 624.93c-1.14 1.138-21.664 13.684-42.315 13.693-20.877.01-41.88-12.542-43.021-13.693L203.122 361.135c-12.923-12.934-12.923-33.885 0-46.807l46.817-46.818z"/></svg></button></div><div class="mermaid-wrapper"><div style="display:flex;align-items:center;justify-content:center;height:96px;" class="mermaid-loading"><span style="--loading-icon: url("data:image/svg+xml;utf8,%3Csvg xmlns='http://www.w3.org/2000/svg' preserveAspectRatio='xMidYMid' viewBox='25 25 50 50'%3E%3CanimateTransform attributeName='transform' type='rotate' dur='2s' keyTimes='0;1' repeatCount='indefinite' values='0;360'%3E%3C/animateTransform%3E%3Ccircle cx='50' cy='50' r='20' fill='none' stroke='currentColor' stroke-width='4' stroke-linecap='round'%3E%3Canimate attributeName='stroke-dasharray' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='1,200;90,200;1,200'%3E%3C/animate%3E%3Canimate attributeName='stroke-dashoffset' dur='1.5s' keyTimes='0;0.5;1' repeatCount='indefinite' values='0;-35px;-125px'%3E%3C/animate%3E%3C/circle%3E%3C/svg%3E");--icon-size: 48px;display: inline-block;width: var(--icon-size);height: var(--icon-size);background-color: currentcolor;-webkit-mask-image: var(--loading-icon);mask-image: var(--loading-icon)"></span></div></div><!--]--><p><strong>数学表示</strong>： 设字母 A-Z 对应数字 0-25。 明文 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo>=</mo><msub><mi>p</mi><mn>0</mn></msub><msub><mi>p</mi><mn>1</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>p</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">P = p_0 p_1 ... p_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 关键词 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mo>=</mo><msub><mi>k</mi><mn>0</mn></msub><msub><mi>k</mi><mn>1</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>k</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">K = k_0 k_1 ... k_{m-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> (长度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>)</p><p>加密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>k</mi><mrow><mi>i</mi><mspace></mspace><mspace width="1em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn></mrow><annotation encoding="application/x-tex">E(p_i) = (p_i + k_{i \pmod m}) \mod 26 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mspace allowbreak mtight"></span><span class="mspace mtight" style="margin-right:0.5204em;"></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mod</span></span></span><span class="mspace mtight" style="margin-right:0.3903em;"></span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span></span></span></span></span></p><p>解密公式：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>c</mi><mi>i</mi></msub><mo>−</mo><msub><mi>k</mi><mrow><mi>i</mi><mspace></mspace><mspace width="1em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace></mspace><mspace width="1em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn></mrow><annotation encoding="application/x-tex">D(c_i) = (c_i - k_{i \pmod m}) \mod 26 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mspace allowbreak mtight"></span><span class="mspace mtight" style="margin-right:0.5204em;"></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mod</span></span></span><span class="mspace mtight" style="margin-right:0.3903em;"></span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:1em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是明文第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 个字母的数字表示，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>i</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">k_{i \pmod m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mspace allowbreak mtight"></span><span class="mspace mtight" style="margin-right:0.5204em;"></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mod</span></span></span><span class="mspace mtight" style="margin-right:0.3903em;"></span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 是关键词循环后对应第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 个字母的数字表示，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">c_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是密文第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 个字母的数字表示。</p><p><strong>示例</strong>： 明文：<code>ATTACKATDAWN</code> 关键词：<code>LEMON</code></p><ol><li><p><strong>关键词循环扩展</strong>： 将关键词 <code>LEMON</code> 循环扩展至与明文等长：<code>LEMONLEMONLE</code></p></li><li><p><strong>明文与关键词按位组合（数字表示）</strong>： 将明文和扩展后的关键词转换为数字 (A=0, B=1, ..., Z=25)。 明文数字: <code>0 19 19 0 2 10 0 19 3 0 22 13</code> 关键词数字: <code>11 4 12 14 13 11 4 12 14 13 11 4</code></p></li><li><p><strong>加密运算</strong>： 对每对明文数字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和关键词数字 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mrow><mi>i</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">k_{i \pmod m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0496em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mspace allowbreak mtight"></span><span class="mspace mtight" style="margin-right:0.5204em;"></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mod</span></span></span><span class="mspace mtight" style="margin-right:0.3903em;"></span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 执行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>p</mi><mi>i</mi></msub><mo>+</mo><msub><mi>k</mi><mrow><mi>i</mi><mspace></mspace><mspace width="0.4444em"></mspace><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"></mspace><mi>m</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn></mrow><annotation encoding="application/x-tex">(p_i + k_{i \pmod m}) \mod 26</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1052em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mspace allowbreak mtight"></span><span class="mspace mtight" style="margin-right:0.5204em;"></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mtight"><span class="mord mathrm mtight">mod</span></span></span><span class="mspace mtight" style="margin-right:0.3903em;"></span><span class="mord mathnormal mtight">m</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span></span></span></span> 运算。 例如：</p><ul><li>第一个字母：明文 A (0) + 关键词 L (11) = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>+</mo><mn>11</mn><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn><mo>=</mo><mn>11</mn><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">(0 + 11) \mod 26 = 11 \rightarrow L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">11</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">11</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span></li><li>第二个字母：明文 T (19) + 关键词 E (4) = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>19</mn><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn><mo>=</mo><mn>23</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(19 + 4) \mod 26 = 23 \rightarrow X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">19</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">23</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></li><li>第三个字母：明文 T (19) + 关键词 M (12) = <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>19</mn><mo>+</mo><mn>12</mn><mo stretchy="false">)</mo><mspace></mspace><mspace width="0.6667em"></mspace><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mtext> </mtext><mtext> </mtext><mn>26</mn><mo>=</mo><mn>5</mn><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">(19 + 12) \mod 26 = 5 \rightarrow F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">19</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">12</span><span class="mclose">)</span><span class="mspace allowbreak"></span><span class="mspace" style="margin-right:0.6667em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord"><span class="mord mathrm">mod</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">26</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> ... 最终密文：<code>LXFOPVEFRNHR</code></li></ul></li></ol><p><strong>安全性分析</strong>： 维吉尼亚密码比单表替换密码更安全，因为它引入了<strong>多表替换</strong>，使得密文的字母频率分布趋于平坦，从而抵抗了简单的频率分析攻击。</p><p>然而，它并非绝对安全，主要弱点在于<strong>关键词的周期性</strong>：</p><ol><li><strong>Kasiski 测试</strong>：通过分析密文中重复出现的字母组，可以推断出关键词的长度。</li><li><strong>频率分析（针对子密码）</strong>：一旦关键词长度确定，密文可以被分成若干个凯撒密码，然后对每个子密码进行频率分析。</li></ol><p><strong>特点</strong>：</p><ul><li>多表替换，比单表替换密码更安全</li><li>引入了关键词的概念，增强了密钥的复杂性</li><li>易受Kasiski测试和频率分析的组合攻击</li><li>在历史上曾被认为是“牢不可破的密码”</li></ul><h2 id="附件" tabindex="-1"><a class="header-anchor" href="#附件"><span>附件：</span></a></h2><p>具体的使用样例代码请参考：<a href="https://gitea.simengweb.com/si-meng-spec/cryptography-example-code" target="_blank" rel="noopener noreferrer">https://gitea.simengweb.com/si-meng-spec/cryptography-example-code</a></p></div><!----><!----><!----></div></main><footer class="vp-doc-footer" data-v-23f6ad98 data-v-7138e2cb><!--[--><!--]--><!----><div class="contributors" aria-label="Contributors" data-v-7138e2cb><span class="contributors-label" data-v-7138e2cb>贡献者: </span><span class="contributors-info" data-v-7138e2cb><!--[--><!--[--><span class="contributor" data-v-7138e2cb>祀梦</span><!----><!--]--><!--]--></span></div><nav class="prev-next" 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