Voronoi Cells of Lattices with Respect to Arbitrary Norms

TL;DR

This paper investigates the geometry and complexity of Voronoi cells of lattices under arbitrary norms, revealing similarities to Euclidean cases for smooth norms but also demonstrating increased combinatorial complexity and implications for lattice algorithms.

Contribution

It extends the understanding of Voronoi cells to arbitrary norms, showing both geometric similarities and increased complexity, and discusses implications for lattice algorithms.

Findings

Voronoi cells are similar to Euclidean case for smooth, strictly convex norms.

The number of Voronoi-relevant vectors can be unbounded for certain norms.

Existing algorithms for Euclidean lattices do not extend efficiently to arbitrary norms.

Abstract

We study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show for strictly convex and smooth norms that the geometry of Voronoi cells of lattices in any dimension is similar to the Euclidean case, i.e., the Voronoi cells are defined by the so-called Voronoi-relevant vectors and the facets of a Voronoi cell are in one-to-one correspondence with these vectors. On the negative side, we show that Voronoi cells are combinatorially much more complicated for arbitrary strictly convex and smooth norms than in the Euclidean case. In particular, we construct a family of three-dimensional lattices whose number of Voronoi-relevant vectors with respect to the $\ell_3$-norm is unbounded. Our result indicates, that the break through single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem in the Euclidean norm cannot be extended to achieve deterministic single exponential time algorithms for lattice problems with respect to arbitrary $\ell_p$-norms. In fact, the algorithm of Micciancio and Voulgaris and its run time analysis crucially depend on the fact that for the Euclidean norm the number of Voronoi-relevant vectors is single exponential in the lattice dimension.