Tensor-based Hardness of the Shortest Vector Problem to within Almost Polynomial Factors

We show that unless NP subset of RTIME(2(poly(logn))), for any epsilon > 0 there is no Polynornial-time algorithm approximating the Shortest Vector Problem (SVP) n-dimensional lattices in the l(p) norm (1 < p < infinity) to within a factor of 2((logn)1-epsilon). This improves the previous best factor of 2((logn)1/2-epsilon) under the same complexity assumption due to Khot [18]. Under the stronger assumption NP Z RSUBEXP, we obtain a hardness factor of a'/ log log n for some c > 0. Our proof starts with SVP instances from [18] that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novel part is in the analysis, where we show that the lattices of [18] behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski [12].