Short Paths on the Voronoi Graph and the Closest Vector Problem with Preprocessing

TL;DR

This paper presents a randomized algorithm for the Closest Vector Problem with preprocessing that improves expected runtime to approximately 2^n, using Voronoi graph paths, and offers insights into geometric-path relationships.

Contribution

It introduces a Las Vegas algorithm with improved expected runtime for CVPP, utilizing a novel randomized path-finding procedure on the Voronoi graph.

Findings

Expected runtime of the algorithm is $ ilde{O}(2^n)$.

Path length to the closest lattice vector is polynomial on average.

Provides an optimal relationship between geometric and path distances on the Voronoi graph.

Abstract

Improving on the Voronoi cell based techniques of Micciancio and Voulgaris (SIAM J. Comp. 13), and Sommer, Feder and Shalvi (SIAM J. Disc. Math. 09), we give a Las Vegas $\tilde{O}(2^n)$ expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the $\tilde{O}(4^n)$ deterministic runtime of the Micciancio Voulgaris algorithm, henceforth MV, for CVPP (which also solves CVP) at the cost of a polynomial amount of randomness (which only affects runtime, not correctness). As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that given the Voronoi relevant vectors of a lattice - the lattice vectors inducing facets of the Voronoi cell - as preprocessing and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the $\tilde{O}(4^n)$ path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors. As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest.