A sublinear time randomized algorithm for coset enumeration in the black box model

Abstract

Coset enumeration is for enumerating the cosets of a subgroup H of a finite index in a group G. We study coset enumeration algorithms by using two random sources to generate random elements in a finite group G and its subgroup H. For a finite set S and a real number c > 0, a random generator R S is a c-random source for S if c· min {Pr[a = R S ())|a ∈ S]}≥ max {Pr[a = R S())|a ∈ S]}. Let c be an arbitrary constant. We present an O (|G|/√|H|(log |G|)3)-time randomized algorithm that, given two respective c-random sources R G for a finite group G and R H for a subgroup H ⊆ G, computes the index t = |G|/|H| and a list of elements a1, a2,...,a t ∈ G such that ai H∩aj H = φ for all i ≠ j, and Ui=1t aiH = G. This algorithm is sublinear time when |H| = Ω((log|G|)6 + ε ) for some constant ε > 0. © 2008 Springer-Verlag Berlin Heidelberg.