On solving systems of random linear disequations

• Authors: Gábor Ivanyos
• Venue: Quantum Information & Computation
• Year: 2008

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.