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(logn)), we show that the problem is hard to approximate within factor 2^{(\log n){1 \mathord{\left/ {\vphantom {1 {2 - \varepsilon }}} \right. \kern-\nulldelimiterspace} {2 - \varepsilon }}} where n is the dimension of the lattice and ∊ > 0 is an arbitrarily small constant. This greatly improves all previous results in ℓ_p norms with 1 < p < ∞. The best results so far gave only a constant factor hardness, namely, 2^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} - \varepsilon by Micciancio [27] and p^{1 - \varepsilon } in high \ell _p norms by Khot [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 \mathord{\left/ {\vphantom {1 {2 - \varepsilon }}} \right. \kern-\nulldelimiterspace} {2 - \varepsilon }}} }.