Baby-Step Giant-Step Algorithms for the Symmetric Group

TL;DR

This paper extends baby-step giant-step algorithms to the symmetric group, enabling efficient computation of permutations as products of smaller sets, with optimal complexity and analysis of randomized collision methods.

Contribution

It develops the first deterministic and randomized algorithms for expressing permutations in S_n as products of small sets, generalizing discrete log techniques to group actions.

Findings

Deterministic algorithms are optimal up to constant factors.

Sets A and B have sizes close to the square root of n!.

Randomized collision algorithms are also analyzed for efficiency.

Abstract

We study discrete logarithms in the setting of group actions. Suppose that $G$ is a group that acts on a set $S$. When $r,s \in S$, a solution $g \in G$ 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 the Shanks baby-step / giant-step procedure for ordinary discrete logarithms. Specifically, we compute two sets $A, B \subseteq S_n$ such that every permutation of $S_n$ can be written as a product $ab$ of elements $a \in A$ and $b \in B$. 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 $\sqrt{n!}$ in size. We also analyze randomized "collision" algorithms for the same problem.