Random permutations from logarithmic signatures

Abstract

A cryptographic system, called pgm, was invented in the late 1970's by S. Magliveras. pgm is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. Logarithmic signatures were initially motivated by C. Sims' bases and strong generators. A logarithmic signature α, for a given group G, induces a mapping \(\hat \alpha\) from Z |G| to G. Hence it would be natural to use logarithmic signatures for generating random elements in a group. In this paper we focus on generating random permutations in the symmetric group S n. Random permutations find applications in design of experiments, simulation, cryptology, voice-encryption etc. Given a logarithmic signature α for S n and a seed s 0, we could efficiently compute the following sequence : \(\hat \alpha\)(s 0), \(\hat \alpha\)(s 0 + 1), ..., \(\hat \alpha\)(s 0 + r - 1) of r permutations. We claim that this sequence behaves like a sequence of random permutations. We undertake statistical tests to substantiate our claim.