Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs

The classical Deuring correspondence provides a roadmap between supersingular elliptic curves and the maximal orders which are isomorphic to their endomorphism rings. Building on this idea, we add the information of a cyclic subgroup of prime order N to supersingular elliptic curves, and prove a generalisation of the Deuring correspondence for these objects. We also study the resulting ell-isogeny graphs supersingular elliptic curve with level-N structure, and the corresponding graphs in the realm of quaternion algebras. The structure of the supersingular elliptic curve ell-isogeny graph underlies the security of a new zero-knowledge proof of isogeny knowledge.