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 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 / giantstep 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.

The alternative Method to Finish Modular Exponentiation and Point Multiplication Processes

“…Step [4], Pohlig-Hellman Algorithm [5] and Index Calculus Algorithm [6]. Moreover, the example algorithms to recover RSA's private key are Wiener's Attack [7], [8], [9] and the new estimated initial value for brute force attack [10].…”