Studying isogeny-based cryptography using lattice geometry

A supersingular isogeny graph is a graph whose vertices are elliptic curves defined over a finite field, and whose edges are certain functions between these curves. These graphs form the foundation of a class of cryptosystems. The most well-known of these, Supersingular Isogney Key Encapsulation (SIKE), was proven to be insecure by Castryck and Decru in 2022, but other systems using these graphs are still being actively considered. In this talk we will introduce these graphs and their connection to lattices in quaternion algebras, and I will report on some results about the geometry of lattices, joint with Eyal Goren, that have implications for the structure and security of supersingular isogeny graphs.