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

We formally define the Lattice Isomorphism Problem for module lattices (module-LIP) in a number field K. This is a generalization of the problem defined by Ducas, Postlethwaite, Pulles, and van Woerden (Asiacrypt 2022), taking into account the arithmetic and algebraic specificity of module lattices from their representation using pseudo-bases. We also provide the corresponding set of algorithmic and theoretical tools for the future study of this problem in a module setting. Our main contribution is 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 cyclotomic fields.