Faster tuple lattice sieving using spherical locality-sensitive filters

TL;DR

This paper introduces improved spherical locality-sensitive filters to enhance tuple lattice sieving, significantly reducing the time complexity for solving the shortest vector problem in high dimensions.

Contribution

It generalizes spherical locality-sensitive filters to optimize space-time tradeoffs in tuple lattice sieving, achieving better theoretical time complexities.

Findings

Triple sieve heuristically solves SVP in time 2^{0.3588d + o(d)}

Triple sieve uses less space and time than current best double sieve

Improves upon previous lattice sieving algorithms with new filtering techniques

Abstract

To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time $2^{0.3588d + o(d)}$. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.