Generating random elements in finite groups

Let G be a finite group of order g. A probability distribution Z on G is called E-uniform if vertical bar Z(x) - 1/g vertical bar <= epsilon/g for each x is an element of G. If x(1), x(2),...,x(m) is a list of elements of G, then the random cube Z(m) := Cubc(x(1),...,x(m)) is the probability distribution where Z(m)(y) is proportional to the number of ways in which y can be written as a product x(1)(epsilon 1)x(2)(epsilon 2)...x(m)(epsilon m) with each epsilon(i) = 0 or 1. Let x(1),...,x(d) be a list of generators for G and consider a sequence of cubes W-k := Cube(x(k)(-1),...,x(1)(-1), x(1),...,x(k)) where, for k > d, x(k) is chosen at random from Wk-1. Then we prove that for each delta > 0 there is a constant K-delta > 0 independent of G such that, with probability at least 1-delta, the distribution W-m is 1/4-uniform when m >= d + K-delta lg vertical bar G vertical bar. 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.