Abstract
We show that several recent "positive" results for lattice problems in the \ell _2 norm also hold in \ell _p norms, for p \ge 2. In particular, for lattices of dimension n: Approximating the shortest and closest vector in the \ell _p norm to within \tilde O (\sqrt n) factors is contained in coNP. Approximating the length of the shortest vector in the \ell _p norm to within \tilde O(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 \ell _p norms by up to \sqrt n factors. Taken together, they can be viewed as a partial converse to recent reductions from the \ell _2 norm to \ell _p 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.