On solving systems of random linear disequations
An important special case of the hidden subgroup problem is equivalent to the hidden shift problem over abelian groups. An efficient solution to the latter problem could serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the hidden shift problem is a reduction to solving systems of certain random disequations in finite abelian groups. By a disequation we mean a constraint of the form $f(x)\neq 0$. In our case, the functions on the left hand side are generalizations of linear functions. The input is a random sample of functions according to a distribution which is up to a constant factor uniform over the "linear" functions $f$ such that $f(u)\neq 0$ for a fixed, although unknown element $u\in A$. The goal is to find $u$, or, more precisely, all the elements $u'\in A$ satisfying the same disequations as $u$. In this paper we give a classical probabilistic algorithm which solves the problem in an abelian $p$-group $A$ in time polynomial in the sample size $N$, where $N=(\log\size{A})^{O(q^2)}$, and $q$ is the exponent of $A$.