Solving the Closest Vector Problem with respect to l_p Norms

Abstract

In this paper, we present a deterministic algorithm for the closest vector problem for all l_p-norms, 1 < p < \infty, and all polyhedral norms, especially for the l_1-norm and the l_{\infty}-norm. We achieve our results by introducing a new lattice problem, the lattice membership problem. We describe a deterministic algorithm for the lattice membership problem, which is a generalization of Lenstra's algorithm for integer programming. We also describe a polynomial time reduction from the closest vector problem to the lattice membership problem. This approach leads to a deterministic algorithm that solves the closest vector problem for all l_p-norms, 1 < p < \infty, in time p log_2 (r)^{O (1)} n^{(5/2+o(1))n} and for all polyhedral norms in time (s log_2 (r))^{O (1)} n^{(2+o(1))n}, where s is the number of constraints defining the polytope and r is an upper bound on the coefficients used to describe the convex body.