On the average hardness of SIVP for module lattices of fixed rank

In joint work with Koen de Boer, Aurel Page, and Benjamin Wesolowski, we study the hardness of the approximate Shortest Independent Vectors Problem (SIVP) for random module lattices. We use here a natural notion of randomness as defined originally by Siegel through Haar measures. By proving a reduction, we show it is essentially as hard as the problem for arbitrary instances. While this was previously known for ideal lattices (those of rank 1), it is the first such result in higher rank. I will give an overview of the reduction and discuss some of the challenges. The work involves deep number theoretic techniques and results, such as the equidistribution of Hecke points, which we study using the spectral theory of automorphic forms. This talk should be accessible to both cryptographers as well as number theorists.