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href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/probability.html" aria-label="概率"><!--[--><!--[--><!--]--><!--]-->概率<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">密码学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/rsa.html" aria-label="RSA"><!--[--><!--[--><!--]--><!--]-->RSA<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/flip_coin.html" aria-label="抛币协议"><!--[--><!--[--><!--]--><!--]-->抛币协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/cryptography/mental_poker.html" aria-label="智能扑克协议"><!--[--><!--[--><!--]--><!--]-->智能扑克协议<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">图形学 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">数学基础 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="从抛币协议到智能合约-二" tabindex="-1"><a class="header-anchor" href="#从抛币协议到智能合约-二"><span>从抛币协议到智能合约(二)</span></a></h1><p>除了抛币协议，加密领域还有一个很有趣的课题和游戏的去中心化直接相关，就是<a href="https://en.wikipedia.org/wiki/Mental_poker" target="_blank" rel="noopener noreferrer">Mental Poker</a>协议，中文一般翻译成“智力扑克”，但我认为这个翻译并不准确，“Mental Poker”一词来源于1979年的一篇论文《<a href="http://people.csail.mit.edu/rivest/ShamirRivestAdleman-MentalPoker.pdf" target="_blank" rel="noopener noreferrer">Mental Poker</a>》，作者是Shamir,Rivest和Adleman（就是发明RSA算法的三位大牛）,里面这么描述这个场景：</p><figure><img src="/images/2018/04/vdice_04.png" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>Mental这个词在这里应该是意念或者思维的意思，和盲棋是对应的，但中文中并没有“意念扑克”这样的词，所以翻译成“盲牌协议”其实更贴切一些。和象棋围棋这样的信息公开游戏不同，扑克游戏大部分都有发牌环节，玩家手中的牌是保密的，所以两位象棋大师完全可以在电话里通过“炮二平五、马八进七”来一局象棋，但扑克牌就不行了，你说手里有一对Ace，我怎么知道你没撒谎呢？<br> 所以Mental Poker协议最关键的就是洗牌和发牌这个流程，这个过程既要保证是随机和公平的，又要保证玩家的手牌的私密性，在这篇论文里给了一个巧妙的方法，先举个形象的例子：</p><blockquote><p>Alice想把一件珍宝通过信使送给Bob，但信使非常不可靠，没有锁进箱子的东西他一定会偷的，并且如果让信使拿到钥匙他也一定会去打开锁的，那在这种情况下Alice和Bob有什么办法安全的把宝物送给对方呢？答案是：Alice用箱子把宝物装进去，然后用自己的锁把箱子锁起来送给Bob，Bob收到箱子后再加上一把自己的锁，把箱子再送回给Alice，Alice收到后打开自己的锁送回给Bob，最后Bob收到箱子之后再打开自己的所，就可以拿到宝物了。</p></blockquote><p>按这个思路换成扑克牌游戏，假设Alice和Bob要玩一局扑克牌游戏，开局的时候他们都要一副牌（52张）中各抽取5张，那么过程就是这样：</p><ol><li>Alice负责洗牌，她首先生成一个长度为52的数组，数组的每个元素表示一张牌，然后用她的加密算法<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>A</mi></msub></math></mjx-assistive-mml></mjx-container>对这些元素进行加密，再把数组顺序打乱（洗牌）之后<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>发送给下Bob</li><li>Bob收到这些牌之后并不认识，因为上面都有Alice的锁，所以他只能随机挑出5张牌，用自己的加密算法之后生成<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>送给Alice</li><li>Alice收到这5张牌之后用自己的解密函数解密，生成<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>���还给Bob</li><li>Bob收到后进行解密，得到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>M</mi></math></mjx-assistive-mml></mjx-container>作为自己的牌</li><li>最后Bob从剩下的47张牌中随机挑出5张送给Alice，Alice直接解密得到<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="4"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>M</mi></math></mjx-assistive-mml></mjx-container>作为她的牌</li><li>发牌结束后双方交换密钥以验证此过程没有作弊</li></ol><hr><p>这个协议中的加密函数需要满足“可交换性”，也就是对一份数据进行多次加密，加密的次序不影响最后的结果：</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>这样在第3步过程中</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D437 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.153em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>D</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>D</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mi>B</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>一般的加密算法并不满足这个特性，但同模的RSA算法可以。 RSA算法其实是生成一对密钥<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mi>e</mi><mo>,</mo><mi>d</mi><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container>，使其满足<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2261"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="4"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mspace style="width:0.444em;"></mjx-mspace><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mi><mjx-mspace style="width:0.333em;"></mjx-mspace><mjx-mi class="mjx-i" space="2"><mjx-c class="mjx-c1D719 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>d</mi><mo>≡</mo><mn>1</mn><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>ϕ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container>，其中<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>是两个大质数的乘积。那么一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>其实可以生成多个密钥对，那么这些同模的RSA算法就满足上面所说的交换律。比如使用同模的两对密钥<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c7B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c7D"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><msub><mi>e</mi><mn>1</mn></msub><mo>,</mo><msub><mi>d</mi><mn>1</mn></msub><mo fence="false" stretchy="false">}</mo><mo>,</mo><mo fence="false" stretchy="false">{</mo><msub><mi>e</mi><mn>2</mn></msub><mo>,</mo><msub><mi>d</mi><mn>2</mn></msub><mo fence="false" stretchy="false">}</mo></math></mjx-assistive-mml></mjx-container>两次对<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container>进行加密</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mstyle><mjx-TeXAtom texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D438 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.026em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-TeXAtom texclass="ORD"><mjx-msup><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4" rspace="4"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4" rspace="4"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4" rspace="4"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-mstyle></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><msub><mi>E</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><msub><mi>e</mi><mn>1</mn></msub></mrow></msup><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>n</mi></mrow><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><msub><mi>e</mi><mn>2</mn></msub></mrow></msup><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>n</mi><mo>=</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><msub><mi>e</mi><mn>1</mn></msub><mo>×</mo><msub><mi>e</mi><mn>2</mn></msub></mrow></msup><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>n</mi></mrow></mstyle></math></mjx-assistive-mml></mjx-container><p>可见加密顺序并不影响结果，这个协议可以很容易扩展到多人扑克游戏，多人之后，第一个玩家<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container>比较特殊，他承担洗牌的任务，然后加密后发给下一个玩家<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>2</mn></msub></math></mjx-assistive-mml></mjx-container>，执行协议中第2到第4步，从收到的牌中随机挑出自己要的牌，加密后送给<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container>解密，然后把剩下的牌发给下一个玩家<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>3</mn></msub></math></mjx-assistive-mml></mjx-container>，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>3</mn></msub></math></mjx-assistive-mml></mjx-container>同样执行第2到4步，以此类推，最后<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-TeXAtom texclass="ORD"><mjx-mi class="mjx-cal mjx-i"><mjx-c class="mjx-c50 TEX-C"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;margin-left:-0.037em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi data-mjx-variant="-tex-calligraphic" mathvariant="script">P</mi></mrow><mn>1</mn></msub></math></mjx-assistive-mml></mjx-container>所需要的牌由最后一个玩家负责抽取。 这个算法公开后，很快就有人发现其中的漏洞，就是RSA算法很可能会泄露扑克牌的信息！如果牌面数字<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math></mjx-assistive-mml></mjx-container>是<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>的二次剩余，那么加密后的信息<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D440 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D452 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4" rspace="4"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c6F"></mjx-c><mjx-c class="mjx-c64"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><msup><mi>M</mi><mi>e</mi></msup><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>也是n的二次剩余，尽管这种泄露微乎其微（可能只有一个bit)，但在某些关键场合也是致命的，所以后来有很多种其他Mental Poker协议出现，有兴趣的朋友可以参考<a href="https://crises-deim.urv.cat/web/docs/publications/theses/jCastella.pdf" target="_blank" rel="noopener noreferrer">这篇文章</a></p><hr><p>有意思的是，尽管目前已经有多种严密的算法来解决扑克牌的去中心化问题，但我们仍然没有看到真的有产品在实际中使用这些协议。这是由于如果想实现真正的去中心化游戏平台，除了刚才说的这些安全协议之外，还需要有一大堆辅助的服务，比如P2P网络基础，玩家匹配，积分，支付等系统，这些系统的复杂度还很难实现去中心化。但区块链和比特币出现之后情况就不同了，2012年，一个使用比特币进行赌博的游戏平台中<a href="https://www.satoshidice.com/" target="_blank" rel="noopener noreferrer">本聪筛子</a>出现了，玩法很简单，玩家给系统提供的比特币钱包转账一定金额的比特币，这笔交易会在区块链网络里会产生一条公开的记录，系统根据这条交易记录的ID计算出一个0到65535之间的幸运数字，系统根据这个数字按照一定的赔率付给玩家比特币，或者收走玩家的比特币。<br> 不难看出，中本聪筛子其实是基于它的中心系统运行的，而以太坊和智能合约出现之后我们距离去中心化又前进了一步，以太坊同样也是以区块链作为基础的去中心化网络，不同的是，它可以支持一种叫做智能合约的特殊数据在链上运行，一份智能合约可以看作是一段代码，比如两个人要打一个赌，那么双方可以把赌金和获胜的判断条件写到智能合约代码中，发布到以太坊上，由系统来判断哪方获胜并自动把赌金转给获胜的一方，由于这个判断的运行是放在整个区块链上运行的，所以可以视作是去中心化的。<br> 比如<a href="https://www.etheroll.com/" target="_blank" rel="noopener noreferrer">etheroll</a>这个平台就是一个利用以太坊智能合约进行赌博的平台，它的玩法和中本聪筛子很类似，也是挑选一个赔率，向指定钱包发送一定数量的以太币开始游戏，然后这个系统会调用发送一份智能合约到以太坊上，合约里会产生一个0到100的随机数，只要你的随机数小于赔率对应的数值，那么系统就会按照赔率付给你对应得以太币。但是，这个随机数如何产生呢？这就要用到以太坊中的“预言机”机制，简单来说，预言机其实就是区块链和显示网络的“连接器”，在普通的智能合约代码里，是不能调用网络函数的，这是由于区块链数据一旦生成，就不能再做任何修改，但预言机提供了一些特殊的合约，其他合约可以通过调用其代码来获取互联网数据，比如随机数、股票数据、天气甚至欧洲杯的结果，在<a href="https://etherscan.io/address/0x048717ea892f23fb0126f00640e2b18072efd9d2#code" target="_blank" rel="noopener noreferrer">Etheroll的智能合约代码</a>里，就是通过调用<a href="www.oraclize.it">www.oraclize.it</a>提供的智能合约，从<a href="random.org">random.org</a>获取的随机数，而这两个机构都是大家公信的机构。</p><figure><img src="/images/2018/04/vdice_05.png" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>但智能合约是有高昂的代价的，根据智能合约的复杂程序，编写者需要支付一定数量的费用(称为Gas)才能提交合约，而且以目前以太坊的机制，即使像扔筛子这样的一次性简单操作，所需要编写的智能合约已经非常复杂了。如果想完成一个去中心化的德州扑克几乎是完全不可能的，就看有没有新的技术平台能够实现这一点了。</p></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/blog/2025/02/MentalPoker01.html" aria-label="从抛币协议到智能合约(一)"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div 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