Baby-step giant-step algorithms for the symmetric group. (March 2018) - Bach, Eric; Sandlund, Bryce
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 r g = s can be thought of as a kind of logarithm. In this paper, we study the case where G = S n and develop analogs to Shanks' baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets A, B ⊆ S n such that every permutation of S n 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.