Generating random elements in finite groups

Abstract

Let G be a finite group of order g. A probability distribution Z on G is called ε-uniform if |Z(x) - 1/g| ≤ ε/g for each x ∈ G. If x1, x2, . . . , xm is a list of elements of G, then the random cube Zm := Cube(x1, . . . , xm) is the probability distribution where Zm(y) is proportional to the number of ways in which y can be written as a product x1ε1x2ε2⋯xmεm with each εi = 0 or 1. Let x1, . . . , xd be a list of generators for G and consider a sequence of cubes Wk := Cube(xk-1, . . . , x 1-1, x1, . . . , xk) where, for k > d, xk is chosen at random from Wk-1. Then we prove that for each δ > 0 there is a constant Kδ > 0 independent of G such that, with probability at least 1-δ, the distribution Wm is 1/4-uniform when m ≥ d + Kδ lg |G|. This justifies a proposed algorithm of Gene Cooperman for constructing random generators for groups. We also consider modifications of this algorithm which may be more suitable in practice.