On solving systems of random linear disequations

An important special case of the hidden subgroup problem is equivalent to the hiddenshift problem over abelian groups. An efficient solution to the latter problem could serveas a building block of quantum hidden subgroup algorithms over solvable groups. Themain idea of a promising approach to the hidden shift problem is a reduction to solvingsystems of certain random disequations in finite abelian groups. By a disequation wemean a constraint of the form f(x) ≠ 0. In our case, the functions on the left handside are generalizations of linear functions. The input is a random sample of functionsaccording to a distribution which is up to a constant factor uniform over the "linear"functions f such that f(u) ≠ 0 for a fixed, although unknown element u ∈ A. The goal isto find u, or, more precisely, all the elements u′ ∈ A satisfying the same disequations asu. In this paper we give a classical probabilistic algorithm which solves the problem inan abelian p-group A in time polynomial in the sample size N, where N = (log |A|)O(q2),and q is the exponent of A.