Hardness of Approximating the Shortest Vector Problem in Lattices

Abstract

Let p > 1 be any fixed real. We show that assuming NP /spl nsube/ RP, it is hard to approximate the shortest vector problem (SVP) in l/sub p/ norm within an arbitrarily large constant factor. Under the stronger assumption NP /spl nsube/ RTIME(2/sup poly(log n)/), we show that the problem is hard to approximate within factor 2/sup log n1/2 - /spl epsi// where n is the dimension of the lattice and /spl epsi/> 0 is an arbitrarily small constant. This greatly improves all previous results in l/sub p/ norms with 1 < p < /spl infin/. The best results so far gave only a constant factor hardness, namely, 2/sup 1/p/ - /spl epsi/ by Micciancio and p/sup 1 - /spl epsi// in high l/sub p/ norms by Khot. We first give a new (randomized) reduction from closest vector problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2/sup log n1/2-/spl epsi//.