## Cryptology ePrint Archive: Report 2009/186

Statistics of Random Permutations and the Cryptanalysis of Periodic Block Ciphers

Nicolas T. Courtois and Gregory V. Bard and 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.

Category / Keywords: secret-key cryptography / Generating Functions, EGF, OGF, Random Permutations, Cycle Structure, Cryptanalysis, Iterations of Permutations, Analytic Combinatorics, Keeloq.

Publication Info: Submitted to a journal, the response is pending.