Voronoi Cells of Lattices with Respect to Arbitrary Norms

Motivated by the deterministic single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem for the Euclidean norm, we study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show that for strictly convex and smooth norms 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 combinatorially Voronoi cells for arbitrary strictly convex and smooth norms are much more complicated 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 `3-norm is unbounded. Since 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, this indicates that the techniques of Micciancio and Voulgaris cannot be extended to achieve deterministic single exponential time algorithms for lattice problems with respect to arbitrary `p-norms.