1、第八届数学建模认证杯网络挑战赛 B题优秀论文1386队第八届“认证杯”数学中国数学建模网络挑战赛 承 诺 书我们仔细阅读了第八届“认证杯”数学中国数学建模网络挑战赛的竞赛规则。 我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问 题。我们知道,抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他 公开的资料(包括网上查到的资料),必须按照规定的参考文献的表述方式在正 文引用处和参考文献中明确列出。我们郑重承诺,严格遵守竞赛规则,以保证竞赛的公正、公平性。如有违反 竞赛规则的行为,我们接受相应处理结果。我
2、们允许数学中国网站()公布论文,以供网友之间学习交流, 数学中国网站以非商业目的的论文交流不需要提前取得我们的同意。我们的参赛队号为:1386 队 参赛队员 (签名) :队员 1: 队员 2: 队员 3:参赛队教练员 (签名): 数模指导小组 参赛队伍组别(例如本科组):本科组第八届“认证杯”数学中国数学建模网络挑战赛 编 号 专 用 页参赛队伍的参赛队号:(请各个参赛队提前填写好):竞赛统一编号(由竞赛组委会送至评委团前编号):竞赛评阅编号(由竞赛评委团评阅前进行编号):2015 年第八届“认证杯”数学中国 数学建模网络挑战赛第一阶段论文题 目 替换式密码自动化破译算法 关 键 词 频率攻击
3、 时间复杂度 单字母加密 自动化破译摘 要:古典密码是密码学的起源,它是由基于字符的密码算法构成,可用并机械操 作实现加解密。目前解密行之有效的方法则是频率分析法,但是利用传统频率分 析法,拥有运算量大,计算时间长,需要过多的人工干预等缺点。所以本文提出 一种创新的全自动解密算法,对单字母加密方法的密文进行解密。模型一:我们对传统频率法进行改进,建立了一种新的全自动化解密模型, 不但能大量减少运算时间,而且可精确高效破译密文。首先我们在模型准备中统计出英文固有的各种频率数据,并建立给定字母长 度 15-24 的长单词库群和字母长度 1-8 字典库群,使模型可实现单词的快速搜索 和匹配。模型一的
4、算法分为三个模块:(1)频率攻击:先统计密文中单字母、 双字母组合频率并分组,再对分组频率进行匹配,从而确定最高频字母 t, h, e。(2)长单词攻击:先将已破译的少量字母代入密文,再从中提取最长的单词代 入相应长度的长单词字典库进行匹配,从而逐步破译更多密钥字母。因单词越长, 数目越少,故匹配成功后正确率很高。(3)动态逐词攻击:先将已破译密钥字 母代入密文,并对密文进行以单词为单位的扫描。再将单词逐个在字典库中进行 匹配,得到相似单词组;其次累加记录每个相似单词中匹配的字母密钥,直至获 取充足信息量;最终选取累计次数最高的匹配为最终密钥,破译出完整密钥。经 多实例统计验证,该解密算法破译
5、大约 5 万字母数的文章,用时仅 2.75s。模型二:为评价模型一中解密算法模型的破译能力,我们从以下两个方面进 行考虑:(1)算法破译的准确性:通过统计分析由全自动解密模型算法破译的 多篇加密文章,得出密钥字母正确个数稳定在 20 个左右,密钥字母的正确率均 高于 80%,全文平均字母正确率稳定在 96%;(2)算法破译密文的时间复杂度: 首先结合算法三个模块程序的复杂度分析得,时间复杂度 F(n)和密文总字母数 n 的关系为 F(n)=(C2+2)n+C1,简记 O(n),其中 C1, C2 是依赖于字典库总词数的 可变常数。其次因为模型算法只需要提取文章中前面部分单词,便可破译出24 个
6、密钥,文章中剩余单词量并不影响算法破译时间,所以文章越长,单位时间内 的破译效率越高。综上所述,该模型能高效准确的解密出密文,算法运行速度快耗时极短,克 服了传统频率分析法耗时长,运算效率不高的缺点。另外,我们还对模型的改进和推广方向以及优缺点进行了讨论。参赛队号: 1386所选题目: B 题AbstractDecryption method is frequency analysis, but the use of traditional frequency analysis method, with large computation, the computation time is lo
7、ng, shortcomings and so on need too much human intervention.So this paper proposes an innovative decryption algorithm, encryption method of single letter the encrypted cryptograph.Model 1: We have to improve the traditional method, a new automatic decryption model is established, not only can substa
8、ntially reduce computing time, and can be accurate and efficient decoding ciphertext. First we have the data needed to measure out the model of the model, the letters length of 15 to 24 long words dictionary library 1-8 of the dictionary, in the model 1 is divided into three steps: (1) the frequency
9、 of cipher attack first, and the statistical frequency of single letters, double letter combinations and grouping, and then grouped frequency matching, analysis of basic to determine the highest frequency of the letter e, he, th, to determine t, h, e, (2) long words on ciphertext attacks, namely in
10、the crack of a small amount of high frequency letters after generation into the ciphertext, extract the longest word in long words dictionary library matching gradually deciphering secret key more letters.Because of the longer the words, the less the number, so after the success of the matching accu
11、racy is higher.(3) the last will be the key letters substitution cipher, and the cipher text with the word for the unit of the scan.Match the words in a dictionary in the library one by one, get the similar words;Accumulated each similar word matching letters secret key again, until the record numbe
12、r of secret key letter = 24, eventually to every secret key letters, select the highest cumulative number of matching as the ultimate secret key, output the complete key.Verified by multi-instance statistics, the decryption algorithm to decipher the number of articles about 50000 letters, it only ne
13、eds 2.75 s.Model 2: To evaluate the model a decryption algorithm decoding ability of the model, we from the following two aspects: (1) considering the accuracy of the algorithm decoding;Tested much of the article, through the Matlab statistic can be concluded that the key letters correct number stab
14、ility in 20 , the key letters correctly about 80%, average letters in full accuracy of about 96% (2) break the time complexity of cipher algorithm.The less as the number of letters, decoding time grows with the increase of the number of letters.While the number of letters after reaching a certain th
15、reshold, decoding time tends to be stable.Because model algorithm only need to extract the partial words in the article the amount of information that can be decoded key.Article the rest of the word does not affect the algorithm decoding time, so the longer the articles, per unit time decoding effic
16、iency is higher.To sum up, this model can efficiently accurately to decrypt the ciphertext, algorithm fast time shortly, overcomes the traditional frequency analysis method time-consuming, the disadvantage of operation efficiency is not high.In addition, we also the improvement and popularization of direction as well as the advantages and disadvantages of the model are discussed.Keywords: Frequency at
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