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**2009/186**

**Title:**

Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers

**Authors:**
Nicolas T. Courtois, Gregory V. Bard, Shaun V. Ault

**Abstract:**

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. This attack is 2^{119.237} times faster than brute-force search. It is hoped that these results will showcase the utility of exponential and ordinary generating functions and will encourage their use in cryptanalytic research.

**ePrint:**
https://eprint.iacr.org/2009/186

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