随机排列的统计与周期分组密码的分析
Statistics of Random Permutations and the Cryptanalysis Of Periodic Block Ciphers
摘要：
分组密码旨在在计算上与适当域和范围的随机排列不可区分。但是随机排列的性质是什么呢？借助于指数生成函数和普通生成函数，我们导出了一系列密码学界感兴趣的集合。这些都遵循强循环结构定理的排列，并在渲染严格的两个攻击Keeloq，在广泛使用的分组密码是有用的。这些攻击以前有启发式近似的成功概率。此外，我们描绘了一个攻击（大致）百万倍迭代的随机排列。特别是，我们创建了一个区别攻击，即密码的迭代次数等于一个特别选择的高度合数是可破解的，但仅仅少一轮就安全得多。然后，我们将其扩展到“Triple-DES”风格构造中的密钥恢复攻击，但使用AES-256并迭代中间密码（大约）一百万倍。希望这些结果将展示指数和普通生成函数的效用，并鼓励它们在密码分析研究中的使用。
英文摘要：
A block cipher is intended to be computationally indistinguishable from a random permutation of appropriate domain and range. But what are the properties of a random permutation? By the aid of exponential and ordinary generating functions, we derive a series of collolaries of interest to the cryptographic community. These follow from the Strong Cycle Structure Theorem of permutations, and are useful in rendering rigorous two attacks on Keeloq, a block cipher in wide-spread use. These attacks formerly had heuristic approximations of their probability of success. Moreover, we delineate an attack against the (roughly) millionth-fold iteration of a random permutation. In particular, we create a distinguishing attack, whereby the iteration of a cipher a number of times equal to a particularly chosen highly-composite number is breakable, but merely one fewer round is considerably more secure. We then extend this to a key-recovery attack in a "Triple-DES" style construction, but using AES-256 and iterating the middle cipher (roughly) a million-fold. It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research.
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