Files
thejinchao.github.io/assets/MentalPoker01.html-CVPMtJle.js
T

2 lines
72 KiB
JavaScript
Raw Blame History

This file contains ambiguous Unicode characters
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
import{_ as a,c as x,a as c,b as m,d as s,o as j}from"./app-Bj0iHonY.js";const t="/images/2018/04/vdice_01.png",e="/images/2018/04/vdice_02.png",n="/images/2018/04/vdice_03.png",i={};function o(r,l){return j(),x("div",null,l[0]||(l[0]=[c('<h1 id="从抛币协议到智能合约-一" tabindex="-1"><a class="header-anchor" href="#从抛币协议到智能合约-一"><span>从抛币协议到智能合约(一)</span></a></h1><p>最近区块链非常火,也有人在讨论游戏和区块链的结合,就目前来看,已有的区块链游戏无非就是两种,一种就是类似于<a href="https://www.cryptokitties.co/" target="_blank" rel="noopener noreferrer">CryptoKitties</a>这样的虚拟资产游戏,还有就是博彩类的游戏,比如<a href="https://www.satoshidice.com/" target="_blank" rel="noopener noreferrer">中本聪筛子</a>,<a href="https://www.vdice.io/" target="_blank" rel="noopener noreferrer">vDice</a>等。</p><figure><img src="'+t+'" alt="" tabindex="0" loading="lazy"><figcaption>这是我当初花了300刀买的一只CryptoKitties,编号448412</figcaption></figure><p>严格来说,这些都不算是真正的游戏,在我看来,区块链游戏起码要做到真正的去中心化,能够实现玩家和玩家之间的互动,并且能提供玩家最基本的游戏乐趣才行。而目前所谓的区块链游戏还远远无法满足这几点,而且现在的区块链被笼罩在非常不好的资本炒作氛围中,所谓的区块链创业大部分都是挂羊头卖狗肉而已。 从技术上讲,游戏的去中心化并不是一件新鲜事,比如加密领域的“抛币协议”可以算是这方面的老祖宗了。早在1981年,数学家Manuel Blum就曾经发表了一篇著名的论文《Coin Flipping by Telephone: A Protocol for Solving Impossible Problems.》,在开篇就提出一个有趣的情景: <img src="'+e+'" alt="" loading="lazy"></p><p>简单的描述就是两个人想通过抛硬币来决定解决他们之间的分歧,那么如何通过电话或者互联网来实现这个行为呢?很显然让其中一个人来随机是不行的,因为他们互不信任,两个人合作呢?比如说双方各自从0和1之间随机一个值,然后把两个值异或作为最终的结果?也不行,比如说Alice随机出0或者1,她把这个数告诉Bob,那么Bob就有机会根据Alice随机到的值操纵最后结果。所以想得到真正公平的结果,抛币协议必须能够满足“抛币入井”的特性,如同把硬币扔进井中,双方只可以去观看而不能改变结果。<br> 一种简单的方法是利用单向函数,比如首先让Alice准备一个随机字符串,其中包含”head”或者”tail”作为抛币结果, 然后把这个字符串的hash值给Bob,让Bob猜是head还是tail,最后Alice把原始的字符串公布以验证。这个协议利用了hash函数是单向函数的特性,由于Bob只收到了hash值,他无法判断原始字符串,所以无法作弊,而Alice如果想抵赖,就需要能够操纵这个hash函数,重新构造出一个字符串使其hash值和发给Bob的那个一摸一样,这也是不可能的(或者说很困难的),在WikiPedia给出了一种这个协议的具体实现。</p><figure><img src="'+n+'" alt="" tabindex="0" loading="lazy"><figcaption></figcaption></figure><p>但Blum给出的协议并不是这样,原因是使用hash函数并不“严肃”,因为很难对其进行严格的安全性分析,尽管Bob无法从hash值反推回原来的字符串,但他很可能发现一些蛛丝马迹,而且靠其中一方直接随机出抛币结果也不公平,因为这需要依赖很高质量且双方都信任的随机函数,Blum的抛币协议可以简单的描述为下面的过程:</p>',7),m("ol",null,[m("li",null,[s("首先Bob准备一个Blum整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n"),m("mo",null,"="),m("mi",null,"p"),m("mi",null,"q")])])]),s(",并且把n发送给Alice")]),m("li",null,[s("Alice随机一个小于"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("且和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("互质的数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s(",计算"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"y"),m("mo",null,"="),m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])]),s(",然后把"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"y")])])]),s("发送给Bob")]),m("li",null,[s("Bob猜"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("的雅可比符号"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mi",null,"x"),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{"data-mjx-texclass":"CLOSE"},")")])])])]),s("是1还是-1,并且把猜测的结果发送给Alice")]),m("li",null,[s("Alice向Bob出示"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])])]),m("li",null,[s("Bob且向Alice出示"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,","),m("mi",null,"q")])])])]),m("li",null,"双方根据Bob是否猜对来决定硬币是正面还是反面,并且检测对方在这个过程中有没有作弊。")],-1),m("p",null,"这个协议利了数论中的一些基本知识,因为并不算复杂,这里先简单介绍一下:",-1),m("hr",null,null,-1),m("h4",{id:"二次剩余-quadratic-residue",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#二次剩余-quadratic-residue"},[m("span",null,[s("二次剩余("),m("a",{href:"https://en.wikipedia.org/wiki/Quadratic_residue",target:"_blank",rel:"noopener noreferrer"},"Quadratic residue"),s(")")])])],-1),m("blockquote",null,[m("p",null,[s("对于整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"q")])])]),s(",如果存在整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("满足"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",null,"≡"),m("mi",null,"q"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s(",那么就称"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"q")])])]),s("是"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的二次剩余")])],-1),m("p",null,[s("比如2就是7的一个二次剩余,因为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"3"),m("mn",null,"2")]),m("mo",null,"≡"),m("mn",null,"2"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"7"),m("mo",{stretchy:"false"},")")])])]),s(",但5就不是7的二次剩余,因为找不到一个数满足"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",null,"≡"),m("mn",null,"5"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"7"),m("mo",{stretchy:"false"},")")])])]),s(",这种情况称5是7的二次非剩余。")],-1),m("hr",null,null,-1),m("h4",{id:"勒让德-雅可比符号",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#勒让德-雅可比符号"},[m("span",null,"勒让德-雅可比符号")])],-1),m("blockquote",null,[m("p",null,[s("设"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p")])])]),s("是一个大于2的质数,"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("是一正整数,那么定义勒让德符号("),m("a",{href:"https://en.wikipedia.org/wiki/Legendre_symbol",target:"_blank",rel:"noopener noreferrer"},"Legendre Symbol"),s(")如下:")]),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",width:"full",style:{"min-width":"19.047em",position:"relative"}},[m("mjx-math",{width:"full",display:"true",class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{width:"full",style:{"min-width":"19.047em"},side:"right"},[m("mjx-table",{style:{width:"auto","min-width":"14.891em",margin:"0 2.078em"}},[m("mjx-itable",{width:"full"},[m("mjx-mlabeledtr",{style:{height:"NaNem"}},[m("mjx-mtd",null,[m("mjx-mstyle",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mrow",{space:"4"},[m("mjx-mo",{class:"mjx-s4"},[m("mjx-c",{class:"mjx-c7B TEX-S4"})]),m("mjx-mtable",{style:{"min-width":"10.31em"},justify:"left"},[m("mjx-table",null,[m("mjx-itable",null,[m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"left","padding-right":"0.5em","padding-bottom":"0.05em"}},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0.5em","padding-bottom":"0.05em"}},[m("mjx-mrow",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"是")]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"的"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"二"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"次"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"剩"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"余")])]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"left","padding-right":"0.5em","padding-top":"0.05em","padding-bottom":"0.05em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0.5em","padding-top":"0.05em","padding-bottom":"0.05em"}},[m("mjx-mrow",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"是")]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"的"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"二"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"次"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"非"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"剩"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"余")])]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"left","padding-right":"0.5em","padding-top":"0.05em"}},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0.5em","padding-top":"0.05em"}},[m("mjx-mrow",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"是")]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-mtext",{class:"mjx-n"},[m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"的"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"倍"),m("mjx-utext",{style:{"font-family":"MJXZERO, serif"},variant:"normal"},"数")])]),m("mjx-tstrut")])])])])]),m("mjx-mo",{class:"mjx-n",style:{"vertical-align":"0.25em"}})])])]),m("mjx-tstrut")])])])]),m("mjx-labels",{style:{width:"19.047em"}},[m("mjx-itable",{align:"right",style:{right:"0"}},[m("mjx-mtr",{style:{height:"3.2em"}},[m("mjx-mtd",{id:"mjx-eqn:1"},[m("mjx-mtext",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"}),m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c29"})]),m("mjx-tstrut",{style:{height:"3.2em","vertical-align":"-1.35em"}})])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true"},[m("mlabeledtr",null,[m("mtd",null,[m("mtext",null,"(1)")]),m("mtd",null,[m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mrow",{"data-mjx-texclass":"ORD"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"p")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"="),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"{"),m("mtable",{rowspacing:".1em",columnspacing:"1em",columnalign:"left left"},[m("mtr",null,[m("mtd",null,[m("mn",null,"1")]),m("mtd",null,[m("mrow",null,[m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"a")]),m("mtext",null,"是"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"p")]),m("mtext",null,"的二次剩余")])])]),m("mtr",null,[m("mtd",null,[m("mo",null,"−"),m("mn",null,"1")]),m("mtd",null,[m("mrow",null,[m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"a")]),m("mtext",null,"是"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"p")]),m("mtext",null,"的二次非剩余")])])]),m("mtr",null,[m("mtd",null,[m("mn",null,"0")]),m("mtd",null,[m("mrow",null,[m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"a")]),m("mtext",null,"是"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"p")]),m("mtext",null,"的倍数")])])])]),m("mo",{"data-mjx-texclass":"CLOSE",fence:"true",stretchy:"true",symmetric:"true"})])])])])])])])])])],-1),m("p",null,[s("勒让德符号一般用"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c28 TEX-S2"})]),m("mjx-mfrac",null,[m("mjx-frac",null,[m("mjx-num",null,[m("mjx-nstrut"),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line"),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut"),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c29 TEX-S2"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"p")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")])])])]),s("或者"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"∣"),m("mi",null,"p"),m("mo",{stretchy:"false"},")")])])]),s("表示,可以直接用欧拉准则("),m("a",{href:"https://en.wikipedia.org/wiki/Euler%27s_criterion",target:"_blank",rel:"noopener noreferrer"},"Euler’s criterion"),s(")计算")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",width:"full",style:{"min-width":"15.729em",position:"relative"}},[m("mjx-math",{width:"full",display:"true",class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{width:"full",style:{"min-width":"15.729em"},side:"right"},[m("mjx-table",{style:{width:"auto","min-width":"11.573em",margin:"0 2.078em"}},[m("mjx-itable",{width:"full"},[m("mjx-mlabeledtr",{style:{height:"NaNem"}},[m("mjx-mtd",null,[m("mjx-mstyle",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])])]),m("mjx-tstrut")])])])]),m("mjx-labels",{style:{width:"15.729em"}},[m("mjx-itable",{align:"right",style:{right:"0"}},[m("mjx-mtr",{style:{height:"2.399em"}},[m("mjx-mtd",{id:"mjx-eqn:2"},[m("mjx-mtext",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"}),m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c29"})]),m("mjx-tstrut",{style:{height:"2.399em","vertical-align":"-0.949em"}})])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true"},[m("mlabeledtr",null,[m("mtd",null,[m("mtext",null,"(2)")]),m("mtd",null,[m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mrow",{"data-mjx-texclass":"ORD"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"p")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"="),m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",{stretchy:"false"},"("),m("mi",null,"p"),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("mn",null,"2")])]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"p"),m("mo",{stretchy:"false"},")")])])])])])])])],-1),m("p",null,"比如",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c39"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c39"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mo",{stretchy:"false"},"("),m("mn",null,"9"),m("mo",null,"∣"),m("mn",null,"13"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("msup",null,[m("mn",null,"9"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",{stretchy:"false"},"("),m("mn",null,"13"),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("mn",null,"2")])]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"13"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])],-1),m("p",null,"所以9是13的二次剩余。 而",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mo",{stretchy:"false"},"("),m("mn",null,"10"),m("mo",null,"∣"),m("mn",null,"13"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("msup",null,[m("mn",null,"10"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",{stretchy:"false"},"("),m("mn",null,"13"),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("mn",null,"2")])]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"13"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",null,"−"),m("mn",null,"1")])])],-1),m("p",null,[s("所以10是13的二次非剩余, 这是由于在同余系统中"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,"−"),m("mn",null,"1"),m("mo",null,"≡"),m("mo",null,"−"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"p"),m("mo",{stretchy:"false"},")")])])])],-1),m("p",null,[s("将勒让德符号扩展到合数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("就是雅可比符号("),m("a",{href:"https://en.wikipedia.org/wiki/Jacobi_symbol",target:"_blank",rel:"noopener noreferrer"},"Jacobi symbol"),s("),同样用"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-sop"},[m("mjx-c",{class:"mjx-c28 TEX-S1"})]),m("mjx-mfrac",null,[m("mjx-frac",null,[m("mjx-num",null,[m("mjx-nstrut"),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line"),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut"),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-sop"},[m("mjx-c",{class:"mjx-c29 TEX-S1"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"n")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")])])])]),s("或者"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("表示。")],-1),m("blockquote",null,[m("p",null,[s("合数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("可以表达成多个质因子的乘积,也就是"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c22EF"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n"),m("mo",null,"="),m("msub",null,[m("mi",null,"p"),m("mn",null,"1")]),m("msub",null,[m("mi",null,"p"),m("mn",null,"2")]),m("mo",null,"⋯"),m("msub",null,[m("mi",null,"p"),m("mi",null,"k")])])])]),s(",那么定义雅可比符号")]),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",width:"full",style:{"min-width":"17.965em",position:"relative"}},[m("mjx-math",{width:"full",display:"true",class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{width:"full",style:{"min-width":"17.965em"},side:"right"},[m("mjx-table",{style:{width:"auto","min-width":"13.809em",margin:"0 2.078em"}},[m("mjx-itable",{width:"full"},[m("mjx-mlabeledtr",{style:{height:"NaNem"}},[m("mjx-mtd",null,[m("mjx-mstyle",null,[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c28 TEX-S2"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-lop"},[m("mjx-c",{class:"mjx-c29 TEX-S2"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mrow",{space:"4"},[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mrow",{space:"2"},[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c22EF"})]),m("mjx-mrow",{space:"2"},[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])])]),m("mjx-tstrut")])])])]),m("mjx-labels",{style:{width:"17.965em"}},[m("mjx-itable",{align:"right",style:{right:"0"}},[m("mjx-mtr",{style:{height:"2.399em"}},[m("mjx-mtd",{id:"mjx-eqn:3"},[m("mjx-mtext",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"}),m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c29"})]),m("mjx-tstrut",{style:{height:"2.399em","vertical-align":"-0.949em"}})])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true"},[m("mlabeledtr",null,[m("mtd",null,[m("mtext",null,"(3)")]),m("mtd",null,[m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"n")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"="),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("msub",null,[m("mi",null,"p"),m("mn",null,"1")])]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("msub",null,[m("mi",null,"p"),m("mn",null,"2")])]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"⋯"),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mi",null,"a"),m("msub",null,[m("mi",null,"p"),m("mi",null,"k")])]),m("mo",{"data-mjx-texclass":"CLOSE"},")")])])])])])])])])],-1),m("p",null,"比如",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mstyle",null,[m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mrow",null,[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c34"}),m("mjx-c",{class:"mjx-c33"})])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mrow",{space:"4"},[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mrow",{space:"2"},[m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c28 TEX-S3"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})])])])])])])]),m("mjx-mo",{class:"mjx-s3"},[m("mjx-c",{class:"mjx-c29 TEX-S3"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mstyle",{displaystyle:"true",scriptlevel:"0"},[m("mrow",{"data-mjx-texclass":"ORD"},[m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mn",null,"7"),m("mn",null,"143")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"="),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mn",null,"7"),m("mn",null,"11")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mrow",{"data-mjx-texclass":"INNER"},[m("mo",{"data-mjx-texclass":"OPEN"},"("),m("mfrac",null,[m("mn",null,"7"),m("mn",null,"13")]),m("mo",{"data-mjx-texclass":"CLOSE"},")")]),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},"("),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])])])],-1),m("p",null,[s("需要注意的是,对于合数,雅可比符号并不能判断二次剩余,比如上面的例子中,尽管"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c7C"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c34"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("mn",null,"7"),m("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),m("mn",null,"143"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])]),s(",但7并不是143的二次剩余")],-1),m("hr",null,null,-1),m("h4",{id:"blum整数-blum-integer",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#blum整数-blum-integer"},[m("span",null,[s("Blum整数("),m("a",{href:"https://en.wikipedia.org/wiki/Blum_integer",target:"_blank",rel:"noopener noreferrer"},"Blum Integer"),s(")")])])],-1),m("blockquote",null,[m("p",null,[s("如果有质数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,","),m("mi",null,"q")])])]),s(",满足"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,"≡"),m("mn",null,"3"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"4"),m("mo",{stretchy:"false"},")"),m("mo",null,","),m("mi",null,"q"),m("mo",null,"≡"),m("mn",null,"3"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"4"),m("mo",{stretchy:"false"},")")])])]),s(",那么称它们的乘积"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mi",null,"q")])])]),s("为Blum整数")])],-1),m("p",null,[s("比如21(3×7),33(3×11),133(7×19)都是Blum整数,Blum整数在加密领域用途非常广泛,因为它有很多有用的特性,比如说"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("是Blum整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的一个二次剩余,那么考察下面的二次剩余方程")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",null,"≡"),m("mi",null,"a"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,[s("首先这个方程有且只有4个解,如果知道n的两个质因子"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mi",null,"q")])])]),s("的情况下,可以在常数时间内计算出这四个解(一般数论书中都有求解二次剩余方程的方法,这里不再详述),但如果不知道质因子"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mi",null,"q")])])]),s("的话,解这个方程非常困难,难度和分解"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的难度相等。"),m("br"),s(" 对于Blum整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s(",这4个解的雅可比符号恰好有两个满足"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])]),s(",另两个解满足"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",null,"−"),m("mn",null,"1")])])]),s(",比如方程"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",null,"≡"),m("mn",null,"4"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"21"),m("mo",{stretchy:"false"},")")])])]),s(",四个解分别是2,19,5,16,其中(2|21)=(19|21)=-1,而(5|21)=(16|21)=1")],-1),m("hr",null,null,-1),m("p",null,[s("回到Blum抛币协议,当知道Blum整数这些特性后,理解这个协议就简单了,这个协议中一共有"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,","),m("mi",null,"q"),m("mo",null,","),m("mi",null,"n"),m("mo",null,","),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y")])])]),s("这几个数,在完成第3步(也就是硬币入井)的时候,Alice知道"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n"),m("mo",null,","),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y")])])]),s(",Bob知道"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,","),m("mi",null,"q"),m("mo",null,","),m("mi",null,"n"),m("mo",null,","),m("mi",null,"y")])])]),m("br"),s(" 对于Bob来说,由于他知道"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的两个质因子,所以他能解出"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"y"),m("mo",null,"≡"),m("msup",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("的4个解,但正如我们上面的分析,这4个解中有两个的雅可比符号是1,另两个是-1,所以他猜对的概率是50%,这也保证了这个协议产生的随机数一定是均匀的。"),m("br"),s(" 那么Alice有可能作弊吗?如果她想作弊,那么她需要准备两个"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s(",满足")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.247em","margin-left":"0"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-spacer",{style:{"margin-top":"0.189em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msubsup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.247em","margin-left":"0"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-spacer",{style:{"margin-top":"0.189em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msubsup",null,[m("mi",null,"x"),m("mn",null,"1"),m("mn",null,"2")]),m("mo",null,"≡"),m("msubsup",null,[m("mi",null,"x"),m("mn",null,"2"),m("mn",null,"2")]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,"并且",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2260"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"x"),m("mn",null,"1")]),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"≠"),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",null,"∣"),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,"那么根据Blum整数的其他特性,可以推导出",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.247em","margin-left":"0"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-spacer",{style:{"margin-top":"0.189em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-msubsup",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.247em","margin-left":"0"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-spacer",{style:{"margin-top":"0.189em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c30"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msubsup",null,[m("mi",null,"x"),m("mn",null,"1"),m("mn",null,"2")]),m("mo",null,"−"),m("msubsup",null,[m("mi",null,"x"),m("mn",null,"2"),m("mn",null,"2")]),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"x"),m("mn",null,"1")]),m("mo",null,"+"),m("msub",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"x"),m("mn",null,"1")]),m("mo",null,"−"),m("msub",null,[m("mi",null,"x"),m("mn",null,"2")]),m("mo",{stretchy:"false"},")"),m("mo",null,"≡"),m("mn",null,"0"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,[s("这相当于分解了"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s(",而大数分解又是著名难题,所以Alice也不可能作弊")],-1)]))}const h=a(i,[["render",o],["__file","MentalPoker01.html.vue"]]),p=JSON.parse('{"path":"/blog/2025/02/MentalPoker01.html","title":"从抛币协议到智能合约(一)","lang":"zh-CN","frontmatter":{"title":"从抛币协议到智能合约(一)","tags":"数学 加密","next":{"text":"从抛币协议到智能合约(二)","link":"MentalPoker02.md"}},"headers":[],"git":{"updatedTime":1740225290000,"contributors":[{"name":"thejinchao","username":"thejinchao","email":"thejinchao@gmail.com","commits":1,"url":"https://github.com/thejinchao"}]},"filePathRelative":"blog/2025/02/MentalPoker01.md"}');export{h as comp,p as data};