The endomorphism ring problem and supersingular isogeny graphs

Supersingular isogeny graphs, which encode supersingular elliptic curves and their isogenies, have recently formed the basis for a number of post-quantum cryptographic protocols. The study of supersingular elliptic curves and their endomorphism rings has a long history and is intimately related to the study of quaternion algebras and their maximal orders. In this thesis, we give a treatment of the theory of quaternion algebras and elliptic curves over finite fields as these relate to supersingular isogeny graphs and computational problems on such graphs, in particular, consolidating and surveying results in the research literature. We also perform some numerical experiments on supersingular isogeny graphs and establish a number of refined upper bounds on supersingular elliptic curves with small non-integer endomorphisms.