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

We show the shortest vector problem in the l₂ norm is NP-hard (for randomized reductions) to approximate within any constant factor less than √2. We also give a deterministic reduction under a reasonable number theoretic conjecture. Analogous results hold in any l_p norm (p⩾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