# 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.

## Keywords

Conjugacy Class Symmetric Group Permutation Group Random Permutation Random Element## Preview

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