Let Pr(x, pi)(alpha ->beta) be the probability of a differential approximation to the n-bit permutation pi, determined with respect to the group (Z(2)**n, x). The probability is determined from the difference table T(x,pi) for which T(x,pi)(alpha,beta) =2**n.Pr(x,pi(alpha ->beta). We show that the distribution of T(x,pi)(alpha,beta) asymptotically follows a Poisson distribution. Let M_(x,pi) = max(alpha, beta not equal to I) T(x,pi)(alpha beta) where I is the identity of (Z(2)**n, x), and define B(n)= ln N**2/ ln ln N**2 where $ = (2**n-1). Our main results are to show that with high probability for a random permutation pi, Pr (2B(n) </= M(x,pi)< 2n) ~ 1, x = +, and Pr( M(x,pi)< 2B(n)) ~ 1, x in #,o, where # and o denote modular addition and modular multiplication. Thus XOR differences admit higher probability approximations for random permutations than differences with respect to # and o. Further, with high probability, the best differential probability for a random 64-bit permutation with respect to XOR differences lies in the
interval [2**-58.6,2**-57].
By: Philip Hawkes, Luke O'Connor
Published in: RZ3018 in 1998
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