Slide reduction, revisited—filling the gaps in svp approximation

Abstract

We show how to generalize Gama and Nguyen’s slide reduction algorithm [STOC ’08] for solving the approximate Shortest Vector Problem over lattices (SVP) to allow for arbitrary block sizes, rather than just block sizes that divide the rank n of the lattice. This leads to significantly better running times for most approximation factors. We accomplish this by combining slide reduction with the DBKZ algorithm of Micciancio and Walter [Eurocrypt ’16]. We also show a different algorithm that works when the block size is quite large—at least half the total rank. This yields the first non-trivial algorithm for sublinear approximation factors. Together with some additional optimizations, these results yield significantly faster provably correct algorithms for δ-approximate SVP for all approximation factors n1/2+ε ≥ δ ≥ nO(1), which is the regime most relevant for cryptography. For the specific values of δ = n1-ε and δ = n2-ε, we improve the exponent in the running time by a factor of 2 and a factor of 1.5 respectively.