Supersingular Isogeny Graphs and Endomorphism Rings: Reductions and Solutions

Abstract

In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the \(\ell \)-isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring’s correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras.