Cryptanalysis of Rank-2 Module-LIP in Totally Real Number Fields

At Asiacrypt 2022, Ducas, Postlethwaite, Pulles, and van Woerden introduced the Lattice Isomorphism Problem for module lattices in a number field $K$ (module-LIP). In this article, we describe an algorithm solving module-LIP for modules of rank $2$ in $K^2$, when $K$ is a totally real number field. Our algorithm exploits the connection between this problem, relative norm equations and the decomposition of algebraic integers as sums of two squares. For a large class of modules (including $\mathcal{O}_K^2$), and a large class of totally real number fields (including the maximal real subfield of cyclotomic fields) it runs in classical polynomial time in the degree of the field and the residue at 1 of the Dedekind zeta function of the field (under reasonable number theoretic assumptions). We provide a proof-of-concept code running over the maximal real subfield of some cyclotomic fields. As a side contribution, we also provide some algorithmic and theoretical tools for the future study of the module-LIP problem.