The Action of a Few Random Permutations on r-Tuples and an Application to Cryptography

We prove that for every r and d≥2 there is a C such that for most choices of d permutations π1, π2, ..., πd of S n , a product of less than C log n of these permutations is needed to map any r-tuple of distinct integers to another r-tuple. We came across this problem while studying a seemingly unrelated cryptographic problem, and use this result in order to show that certain cryptographic devices using permutation automata are highly insecure. The proof techniques we develop here give more general results, and constitute a first step towards the study of expansion properties of random Cayley graphs over the symmetric group, whose relevance to theoretical computer science is well-known (see [B&al90]).

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