Lattice reduction for modules, or how to reduce modulesvp to modulesvp

Abstract

This is the extended abstract of [MS20]. See the full version at eprint:2019/1142. We show how to generalize lattice reduction algorithms to module lattices. Specifically, we reduce γ-approximate ModuleSVP over module lattices with rank k ≥ 2 to γ'-approximate ModuleSVP over module lattices with rank 2 ≤ β ≤ k. To do so, we modify the celebrated slide-reduction algorithm of Gama and Nguyen to work with module filtrations, a high-dimensional generalization of the (Z-)basis of a lattice. The particular value of γ that we achieve depends on the underlying number field K, the order (Formula Presented), and the embedding (as well as, of course, k and β). However, for reasonable choices of these parameters, the resulting value of γ is surprisingly close to the one achieved by “plain” lattice reduction algorithms, which require an arbitrary SVP oracle in the same dimension. In other words, we show that ModuleSVP oracles are nearly as useful as SVP oracles for solving higher-rank instances of approximate ModuleSVP. Our result generalizes the recent independent result of Lee, Pellet-Mary, Stehlé, and Wallet, which works in the important special case when β = 2 and R = OK is the ring of integers of K under the canonical embedding, while our reduction works. Indeed, at a high level our reduction can be thought of as a generalization of theirs in roughly the same way that block reduction generalizes LLL reduction. In this extended abstract, we present a special case of the more general result to appear in the full version [MS20].