import{_ as m,c as e,a as n,e as l,b as s,r as p,o as i}from"./app-CsTq_I9B.js";const r={};function c(o,a){const t=p("Mermaid");return i(),e("div",null,[a[0]||(a[0]=n('<h2 id="_1-密码学的定义" tabindex="-1"><a class="header-anchor" href="#_1-密码学的定义"><span>1. 密码学的定义</span></a></h2><h3 id="_1-1-基本概念" tabindex="-1"><a class="header-anchor" href="#_1-1-基本概念"><span>1.1 基本概念</span></a></h3><p><strong>密码学（Cryptography）</strong> 是一门研究信息安全的学科，主要关注如何在不安全的环境中实现安全通信。其核心是通过数学方法对信息进行变换，使得只有授权方能够理解信息内容。</p><h3 id="_1-2-核心目标" tabindex="-1"><a class="header-anchor" href="#_1-2-核心目标"><span>1.2 核心目标</span></a></h3><p>密码学追求以下四个主要安全目标：</p><ul><li><strong>机密性（Confidentiality）</strong>：确保信息只能被授权的人访问</li><li><strong>完整性（Integrity）</strong>：确保信息在传输过程中不被篡改</li><li><strong>认证性（Authentication）</strong>：确认通信双方的身份真实性</li><li><strong>不可否认性（Non-repudiation）</strong>：防止发送方事后否认发送过信息</li></ul><h3 id="_1-3-重要作用" tabindex="-1"><a class="header-anchor" href="#_1-3-重要作用"><span>1.3 重要作用</span></a></h3><p>密码学在现代信息安全中扮演着至关重要的角色：</p><ul><li>保护个人隐私和商业机密</li><li>确保金融交易的安全性</li><li>维护国家安全和军事通信</li><li>支撑互联网基础设施的安全运行</li></ul><h3 id="_1-4-主要应用场景" tabindex="-1"><a class="header-anchor" href="#_1-4-主要应用场景"><span>1.4 主要应用场景</span></a></h3><ul><li><strong>网络安全</strong>：HTTPS、VPN、SSL/TLS协议</li><li><strong>数字身份认证</strong>：数字证书、数字签名、双因素认证</li><li><strong>区块链技术</strong>：加密货币、智能合约、分布式账本</li><li><strong>移动通信</strong>：SIM卡加密、移动支付安全</li><li><strong>物联网安全</strong>：设备身份认证、数据传输加密</li></ul><h3 id="_1-5-基础概念与术语-入门" tabindex="-1"><a class="header-anchor" href="#_1-5-基础概念与术语-入门"><span>1.5 基础概念与术语（入门）</span></a></h3><p>为方便初学者快速建立直觉，先认识密码学中最核心的几个概念：</p><p><strong>明文（Plaintext）与密文（Ciphertext）</strong></p><ul><li>明文：未加密的原始消息，例如“HELLO”。</li><li>密文：加密后的消息，人类或未授权系统难以直接理解。</li></ul><p><strong>加密（Encryption）与解密（Decryption）</strong></p><ul><li>加密：用密钥将明文转换为密文，记为：</li></ul><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>C</mi><mo>=</mo><msub><mi>E</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = E_k(P) </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;">C</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"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span></span></span></span></span></p><ul><li>解密：用密钥将密文还原为明文，记为：</li></ul><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>P</mi><mo>=</mo><msub><mi>D</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P = D_k(C) </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:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</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><mi>P</mi></mrow><annotation encoding="application/x-tex">P</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></span></span> 表示明文，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</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;">C</span></span></span></span> 表示密文，<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> 表示密钥，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</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:0.05764em;">E</span></span></span></span> 为加密算法，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</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:0.02778em;">D</span></span></span></span> 为解密算法。</p><p><strong>密钥（Key）：对称密钥 vs 非对称密钥</strong></p><ul><li>对称密钥：加密和解密使用相同的密钥，速度快，但密钥分发与管理是难点。</li><li>非对称密钥（公钥密码）：加密使用“公钥”，解密使用“私钥”，便于密钥分发，还能支持数字签名。</li></ul><p>对称加密流程示意（同一把密钥）：</p>',24)),l(t,{code:"eJxLy8kvT85ILCpR8AniUlAIjn7aP/FlQ+OLhtZYBV1dhSd79z+fsuJp68Ynu1Y/Xd/2ctJSBW+Fp10LgGygtJ2Cc7QGkPlsWrtmLFC3M04tL5YvhmkJin7Wt/TZlG0gK7gAfgE/qw=="}),a[1]||(a[1]=s("p",null,"非对称加密流程示意（公钥/私钥）：",-1)),l(t,{code:"eJxLy8kvT85ILCpR8AniUlAIjn7aP/FlQ+OLhtZYBV1dhSd79z+fsuJZ39JnU7YBxZ62rnk5aenTrgVP17cBpe0UnKM1gMxn09o1Y4G6nbFoeb68EajlxfLFMC1B0XC5WC4A+5BCKg=="}),a[2]||(a[2]=s("p",null,"在典型的 RSA 公钥体制中，还可以用一个简洁的数学表达式表示加解密：",-1)),a[3]||(a[3]=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mtable",{rowspacing:"0.25em",columnalign:"right left",columnspacing:"0em"},[s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"c")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("msup",null,[s("mi",null,"m"),s("mi",null,"e")]),s("mtext",null," "),s("mo",{lspace:"0.22em",rspace:"0.22em"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"d")])]),s("mtext",null," "),s("mi",null,"n"),s("mo",{separator:"true"},",")])])])]),s("mtr",null,[s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mi",null,"m")])]),s("mtd",null,[s("mstyle",{scriptlevel:"0",displaystyle:"true"},[s("mrow",null,[s("mrow"),s("mo",null,"="),s("msup",null,[s("mi",null,"c"),s("mi",null,"d")]),s("mtext",null," "),s("mo",{lspace:"0.22em",rspace:"0.22em"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"d")])]),s("mtext",null," "),s("mi",null,"n"),s("mo",{separator:"true"},",")])])])])]),s("annotation",{encoding:"application/x-tex"},"\\begin{aligned} c &= m^{e} \\bmod n,\\\\ m &= c^{d} \\bmod n, \\end{aligned} ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"3.0591em","vertical-align":"-1.2796em"}}),s("span",{class:"mord"},[s("span",{class:"mtable"},[s("span",{class:"col-align-r"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7796em"}},[s("span",{style:{top:"-3.9396em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c")])]),s("span",{style:{top:"-2.3804em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2796em"}},[s("span")])])])]),s("span",{class:"col-align-l"},[s("span",{class:"vlist-t vlist-t2"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.7796em"}},[s("span",{style:{top:"-3.9396em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mod")])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mpunct"},",")])]),s("span",{style:{top:"-2.3804em"}},[s("span",{class:"pstrut",style:{height:"3em"}}),s("span",{class:"mord"},[s("span",{class:"mord"}),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8991em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mod")])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mpunct"},",")])])]),s("span",{class:"vlist-s"},"​")]),s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"1.2796em"}},[s("span")])])])])])])])])])])],-1)),a[4]||(a[4]=n('<p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>e</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e, n)</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 mathnormal">e</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 为公钥，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d, n)</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 mathnormal">d</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 为私钥，<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> 为明文，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</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">c</span></span></span></span> 为密文。</p><p><strong>常见攻击模型简介（只需直观理解）</strong></p><ul><li>唯密文攻击（COA）：攻击者只有密文，尝试恢复明文或密钥。</li><li>已知明文攻击（KPA）：攻击者拥有部分“明文-密文”对，用于分析算法或密钥。</li><li>选择明文攻击（CPA）：攻击者可选择明文并获取其密文，用于推断密钥或算法结构。</li><li>选择密文攻击（CCA）：攻击者可选择密文并得到其解密结果，进一步分析系统弱点。</li></ul><p>直观结论：设计良好的现代密码系统，应当在这些攻击模型下仍保持安全（在合理的参数与假设下）。</p><h2 id="_2-密码学历史简述" tabindex="-1"><a class="header-anchor" href="#_2-密码学历史简述"><span>2. 密码学历史简述</span></a></h2><h3 id="_2-1-古代密码学-公元前-15世纪" tabindex="-1"><a class="header-anchor" href="#_2-1-古代密码学-公元前-15世纪"><span>2.1 古代密码学（公元前-15世纪）</span></a></h3><p><strong>凯撒密码（Caesar Cipher）</strong></p><ul><li>时间：公元前1世纪</li><li>原理：字母移位加密</li><li>示例：将字母向后移动3位，A→D，B→E</li></ul><p><strong>斯巴达密码棒（Scytale）</strong></p><ul><li>时间：公元前5世纪</li><li>原理：缠绕在特定直径木棒上的皮条</li></ul><p>古典密码简述：</p><ul><li>核心思路：替换或移位（重新排列）字符。</li><li>代表示例：凯撒（替换）、栅栏（移位）、维吉尼亚（多表替换）。</li><li>直觉目标：混淆结构、增加猜测难度；但易受频率分析。</li></ul><h3 id="_2-2-文艺复兴时期-15-18世纪" tabindex="-1"><a class="header-anchor" href="#_2-2-文艺复兴时期-15-18世纪"><span>2.2 文艺复兴时期（15-18世纪）</span></a></h3><p><strong>维吉尼亚密码（Vigenère Cipher）</strong></p><ul><li>时间：16世纪</li><li>原理：多表替换密码</li><li>特点：比单表替换更安全</li></ul><p><strong>博福特密码（Beaufort Cipher）</strong></p><ul><li>时间：18世纪</li><li>原理：改进的维吉尼亚密码</li></ul><h3 id="_2-3-近代密码学-19-20世纪中期" tabindex="-1"><a class="header-anchor" href="#_2-3-近代密码学-19-20世纪中期"><span>2.3 近代密码学（19-20世纪中期）</span></a></h3><p><strong>恩尼格玛密码机（Enigma）</strong></p><ul><li>时间：二战时期</li><li>原理：机械转子密码机</li><li>重要性：推动了现代密码分析的发展</li></ul><p><strong>香农的信息论</strong></p><ul><li>时间：1949年</li><li>贡献：为密码学奠定了数学理论基础</li></ul><h3 id="_2-4-现代密码学-1970年代至今" tabindex="-1"><a class="header-anchor" href="#_2-4-现代密码学-1970年代至今"><span>2.4 现代密码学（1970年代至今）</span></a></h3><p><strong>DES算法</strong></p><ul><li>时间：1977年</li><li>意义：第一个公开的加密标准</li></ul><p><strong>RSA算法</strong></p><ul><li>时间：1977年</li><li>意义：第一个实用的公钥密码系统</li></ul><p><strong>AES算法</strong></p><ul><li>时间：2001年</li><li>意义：取代DES的新一代加密标准</li></ul><p>现代密码简述：</p><ul><li>对称加密：同一密钥加解密，适合大量数据（示例：AES/DES/3DES）。</li></ul><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>C</mi><mo>=</mo><msub><mi>E</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi>P</mi><mo>=</mo><msub><mi>D</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = E_k(P), \\quad P = D_k(C) </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;">C</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"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></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:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</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="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose">)</span></span></span></span></span></p><ul><li>非对称加密：公钥加密、私钥解密，便于密钥分发与数字签名（示例：RSA/ECC）。</li></ul>',33)),a[5]||(a[5]=s("p",{class:"katex-block"},[s("span",{class:"katex-display"},[s("span",{class:"katex"},[s("span",{class:"katex-mathml"},[s("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[s("semantics",null,[s("mrow",null,[s("mi",null,"c"),s("mo",null,"="),s("msup",null,[s("mi",null,"m"),s("mi",null,"e")]),s("mtext",null," "),s("mo",{lspace:"0.22em",rspace:"0.22em"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"d")])]),s("mtext",null," "),s("mi",null,"n"),s("mo",{separator:"true"},","),s("mspace",{width:"1em"}),s("mi",null,"m"),s("mo",null,"="),s("msup",null,[s("mi",null,"c"),s("mi",null,"d")]),s("mtext",null," "),s("mo",{lspace:"0.22em",rspace:"0.22em"},[s("mrow",null,[s("mi",{mathvariant:"normal"},"m"),s("mi",{mathvariant:"normal"},"o"),s("mi",{mathvariant:"normal"},"d")])]),s("mtext",null," "),s("mi",null,"n")]),s("annotation",{encoding:"application/x-tex"},"c = m^{e} \\bmod n, \\quad m = c^{d} \\bmod n ")])])]),s("span",{class:"katex-html","aria-hidden":"true"},[s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"c"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.7144em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"m"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.7144em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"e")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mod")])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.625em","vertical-align":"-0.1944em"}}),s("span",{class:"mord mathnormal"},"n"),s("span",{class:"mpunct"},","),s("span",{class:"mspace",style:{"margin-right":"1em"}}),s("span",{class:"mspace",style:{"margin-right":"0.1667em"}}),s("span",{class:"mord mathnormal"},"m"),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}}),s("span",{class:"mrel"},"="),s("span",{class:"mspace",style:{"margin-right":"0.2778em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.8991em"}}),s("span",{class:"mord"},[s("span",{class:"mord mathnormal"},"c"),s("span",{class:"msupsub"},[s("span",{class:"vlist-t"},[s("span",{class:"vlist-r"},[s("span",{class:"vlist",style:{height:"0.8991em"}},[s("span",{style:{top:"-3.113em","margin-right":"0.05em"}},[s("span",{class:"pstrut",style:{height:"2.7em"}}),s("span",{class:"sizing reset-size6 size3 mtight"},[s("span",{class:"mord mtight"},[s("span",{class:"mord mathnormal mtight"},"d")])])])])])])])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}}),s("span",{class:"mbin"},[s("span",{class:"mord"},[s("span",{class:"mord mathrm"},"mod")])]),s("span",{class:"mspace",style:{"margin-right":"0.0556em"}}),s("span",{class:"mspace",style:{"margin-right":"0.2222em"}})]),s("span",{class:"base"},[s("span",{class:"strut",style:{height:"0.4306em"}}),s("span",{class:"mord mathnormal"},"n")])])])])],-1)),a[6]||(a[6]=n(`<ul><li>密钥交换：Diffie–Hellman 在不安全信道建立共享密钥。</li><li>数字签名：私钥签名、公钥验证，保障真实性与不可否认性。</li></ul><h3 id="_2-5-关键历史时间线" tabindex="-1"><a class="header-anchor" href="#_2-5-关键历史时间线"><span>2.5 关键历史时间线</span></a></h3><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>公元前5世纪：斯巴达密码棒</span></span>
<span class="line"><span>公元前1世纪：凯撒密码</span></span>
<span class="line"><span>16世纪：维吉尼亚密码</span></span>
<span class="line"><span>1918年：一次一密密码本</span></span>
<span class="line"><span>1949年：香农信息论</span></span>
<span class="line"><span>1977年：DES和RSA算法</span></span>
<span class="line"><span>2001年：AES标准</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 class="line-number"></div><div class="line-number"></div><div class="line-number"></div><div class="line-number"></div></div></div><h2 id="总结" tabindex="-1"><a class="header-anchor" href="#总结"><span>总结</span></a></h2><p>密码学作为信息安全的基石，经历了从简单替换到复杂数学算法的漫长发展历程。现代密码学建立在严格的数学基础之上，通过对称加密、非对称加密等多种技术手段，为数字世界提供了可靠的安全保障。</p><p>理解密码学的基本原理和分类，有助于我们更好地应用这些技术来保护信息安全，同时也为深入学习更高级的密码学概念奠定基础。</p><p>本篇笔记的所有代码开源于：<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>`,7))])}const g=m(r,[["render",c]]),d=JSON.parse('{"path":"/theory/cryptography/","title":"密码学基础","lang":"zh-CN","frontmatter":{"title":"密码学基础","createTime":"2025/10/27 10:38:57","permalink":"/theory/cryptography/"},"readingTime":{"minutes":5.34,"words":1603},"git":{"createdTime":1761564580000,"updatedTime":1761716737000,"contributors":[{"name":"祀梦","username":"","email":"3501646051@qq.com","commits":2,"avatar":"https://gravatar.com/avatar/6406a81eeddc359cf3d3ce018797689fc6d014ff06215c27d0210b42e8f5a8ab?d=retro"}]},"filePathRelative":"notes/theory/cryptography/README.md","headers":[]}');export{g as comp,d as data};