Sieving for Shortest Vectors in Lattices Using Angular Locality-Sensitive Hashing

Abstract

By replacing the brute-force list search in sieving algorithms with Charikar’s angular locality-sensitive hashing (LSH) method, we get both theoretical and practical speedups for solving the shortest vector problem (SVP) on lattices. Combining angular LSH with a variant of Nguyen and Vidick’s heuristic sieve algorithm, we obtain heuristic time and space complexities for solving SVP of \(2^{0.3366n + o(n)}\) and \(2^{0.2075n + o(n)}\) respectively, while combining the same hash family with Micciancio and Voulgaris’ GaussSieve algorithm leads to an algorithm with (conjectured) heuristic time and space complexities of \(2^{0.3366n + o(n)}\). Experiments with the GaussSieve-variant show that in moderate dimensions the proposed HashSieve algorithm already outperforms the GaussSieve, and the practical increase in the space complexity is much smaller than the asymptotic bounds suggest, and can be further reduced with probing. Extrapolating to higher dimensions, we estimate that a fully optimized and parallelized implementation of the GaussSieve-based HashSieve algorithm might need a few core years to solve SVP in dimension 130 or even 140.