Application of automorphic forms to lattice problems

Abstract

Abstract In this article, we propose a new approach to the study of lattice problems used in cryptography. We specifically focus on module lattices of a fixed rank over some number field. An essential question is the hardness of certain computational problems on such module lattices, as the additional structure may allow exploitation. The fundamental insight is the fact that the collection of those lattices are quotients of algebraic manifolds by arithmetic subgroups. Functions in these spaces are studied in mathematics as part of the number theory. In particular, those form a module over the Hecke algebra associated with the general linear group. We use results on these function spaces to define a class of distributions on the space of lattices. Using the Hecke algebra, we define Hecke operators associated with collections of prime ideals of the number field and show a criterion on distributions to converge to the uniform distribution, if the Hecke operators are applied to the chosen distribution. Our approach is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski (CRYPTO’20) on self-reduction of ideal lattices via Arakelov divisors.