New (And Old) Proof Systems For Lattice Problems

We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem GapSPP of approximating the epsilon-smoothing parameter (for some epsilon < 1/2) of an n-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and GapSPP has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that GapSPP admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly root n. Specifically:- There is a noninteractive SZK proof for O(log(n)root log(1/epsilon))-approximate GapSPP. Moreover, for any negligible e and a larger approximation factor <(O)over tilde> (root n log(1/epsilon)), there is such a proof with an efficient prover.-There is an (interactive) SZK proof with an efficient prover for O(log n + root log(1/epsilon)/log n)-approximate coGapSPP. We show this by proving that O(log n)-approximate GapSPP is in coNP.In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an O(v root n) factor, improving upon the prior best factor of omega(root n log n).