Limits on the Hardness of Lattice Problems in ell _p Norms

Abstract

We show that several recent "positive" results for lattice problems in the l2 norm also hold in lp norms, for p>2. In particular, for lattices of dimension n: (i) approximating the shortest and closest vector in the lp norm to within O macr(radicn) factors is contained in coNP, (ii) approximating the length of the shortest vector in the lp norm to within O breve(n) factors reduces to the average-case problems studied in related works (Ajtai, STOC 1996; Micciancio and Regev, FOCS 2004; Regev, STOC 2005). These results improve upon prior understanding of lp norms by up to radicn factors. Taken together, they can be viewed as a partial converse to recent reductions from the l2 norm to lp norms (Regev and Rosen, STOC 2006). One of our main technical contributions is a very general analysis of Gaussian distributions over lattices, which may be of independent interest. Our proofs employ analytical techniques of Banaszczyk which, to our knowledge, have yet to be exploited in computer science.