Baby-step giant-step algorithms for the symmetric group

Abstract

We study discrete logarithms in the setting of group actions. Suppose that G is a group that acts on a set S. When r,s∈S, a solution g∈G to rg=s can be thought of as a kind of logarithm. In this paper, we study the case where G=Sn and develop analogs to Shanks' baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets A,B⊆Sn such that every permutation of Sn can be written as a product ab of elements a∈A and b∈B. Our deterministic procedure is optimal up to constant factors, in the sense that A and B can be computed in optimal asymptotic complexity, and |A| and |B| are a small constant from n! in size. We also analyze randomized “collision” algorithms for the same problem.