Approximating the SVP to within a factor (1+1/dimε) is NP-hard under randomized reductions

Abstract

Recently Ajtai showed that to approximate the shortest lattice vector in the l2-norm within a factor (1+2-dim(k)), for a sufficiently large constant k, is NP-hard under randomized reductions. We improve this result to show that to approximate a shortest lattice vector within a factor (1+dim-ε), for any ε>0, is NP-hard under randomized reductions. Our proof also works for arbitrary lp-norms, 1≤p<∞.