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 \(\textsf {GapSPP}\) of approximating the \(\varepsilon \) -smoothing parameter (for some \(\varepsilon < 1/2\)) of an n-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and \(\textsf {GapSPP}\) has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions. We show that \(\textsf {GapSPP}\) admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly \(\sqrt{n}\). Specifically: There is a noninteractive SZK proof for \(O(\log (n) \sqrt{\log (1/\varepsilon )})\)-approximate \(\textsf {GapSPP}\). Moreover, for any negligible \(\varepsilon \) and a larger approximation factor \(\widetilde{O}(\sqrt{n \log (1/\varepsilon )})\), there is such a proof with an efficient prover. There is an (interactive) SZK proof with an efficient prover for \(O(\log n + \sqrt{\log (1/\varepsilon )/\log n})\)-approximate coGapSPP. We show this by proving that \(O(\log n)\)-approximate \(\textsf {GapSPP}\) is in \(\mathsf {coNP} \).