Tensor-based Hardness of the Shortest Vector Problem to within Almost Polynomial Factors
Résumé
We show that unless NP⊆RTIME(2poly(logn)), there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices in the ℓp norm (1≤p<∞) to within a factor of 2(logn)1−ε for any ε>0. This improves the previous best factor of 2(logn)1/2−ε under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption NP⊈RSUBEXP, we obtain a hardness factor of nc/loglogn for some c>0.
Our proof starts with Khot's SVP instances 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 Khot 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.