International Association
for Cryptologic Research

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 O_K^2, it runs in classical polynomial time (under reasonable number theoretic assumptions). We provide a proof-of-concept code running over the maximal real subfield of cyclotomic fields.