Average hardness of SIVP for module lattices of fixed rank

Abstract

The problem of finding short vectors in Euclidean lattices is a central hard problem in complexity theory. The case of module lattices (i.e., lattices which are also modules over a number ring) is of particular interest for cryptography and computational number theory. The hardness of finding short vectors in the asymptotic regime where the rank (as a module) is fixed is supporting the security of quantum-resistant cryptographic standards such as ML-DSA and ML-KEM. In this article we prove the average-case hardness of this problem for uniformly random module lattices (with respect to the natural invariant measure on the space of module lattices of any fixed rank). More specifically, we prove a polynomial-time worst-case to average-case self-reduction for the approximate Shortest Independent Vector Problem ($\gamma$-SIVP) where the average case is the (discretized) uniform distribution over module lattices, with a polynomially-bounded loss in the approximation factor, assuming the Extended Riemann Hypothesis. This result was previously known only in the rank-1 case (so-called ideal lattices). That proof critically relied on the fact that the space of ideal lattices is a compact group. In higher rank, the space is neither compact nor a group. Our main tool to overcome the resulting challenges is the theory of automorphic forms, which we use to prove a new quantitative rapid equidistribution result for random walks in the space of module lattices.