The shortest vector in a lattice is hard to approximate to within some constant

Abstract

We show the shortest vector problem in the l/sub 2/ norm is NP-hard (for randomized reductions) to approximate within any constant factor less than /spl radic/2. We also give a deterministic reduction under a reasonable number theoretic conjecture. Analogous results hold in any l/sub p/ norm (p/spl ges/1). In proving our NP-hardness result, we give an alternative construction satisfying Ajtai's probabilistic variant of Sauer's lemma, that greatly simplifies Ajtai's original proof.