Paper 2026/1805

On Module Lattices with Galois-Symmetries: What You See Is Not What You Get

Abstract

This paper deals with the hardness of finding short vectors in module lattices. Let $K$ be a number field of degree $d$ and $\mathcal{O}_K$ its ring of integers. We show that if a module lattice $M$ of rank $n$ in $\mathcal{O}_K^n$ has some Galois-symmetries, namely if it is fixed coordinate-wise (as a set) by a group $G$ of automorphisms of $K$, then $M$ can actually be seen as a module of rank~$n$ over a subfield~$K'$ of $K$ ($K'$ is the fixed-field of $G$), whose degree is $|G|$ times smaller than the degree of $K$. When one wants to find short vectors in $M$, this translates into the observation that the module lattice $M$, which is a priori a lattice of rank $n d$ can in fact be seen as a lattice of rank only $n d / |G|$. Hence, finding short vectors in $M$ is easier than what one could have expected by forgetting about the algebraic structure of $M$. This result is a generalization of a similar result by Boudgoust, Gachon and Pellet-Mary (Crypto'22), which was restricted to ideal lattices (i.e., modules of rank $1$).