Hardness of approximating the shortest vector problem in lattices

Let p > 1 be any fixed real. We show that assuming NP ⊂ RP, it is hard to approximate the Shortest Vector Problem (SVP) in l _p norm within an arbitrarily large constant factor. Under the stronger assumption NP ⊂ RTIME(2 ^{poly((lop n)}), we show that the problem is hard to approximate within factor 2(log n) ^1/2-ε where n is the dimension of the lattice and ε > 0 is an arbitrarily small constant. This greatly improves all previous results in _p norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 2 ^1/p - ε by Micciancio [27] and ^1/ε in high l _p norms by Knot [20]. 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(log n) ^1/2-ε.