A supersingular isogeny graph is a graph whose vertices are supersingular elliptic curves over $\overline{\mathbb{F}}_p$ (where $p$ is typically a large prime in our context), and whose edges represent isogenies of degree $\ell$ (typically a small prime). Hard problems concerning pathfinding in supersingular isogeny graphs form a basis for post-quantum isogeny-based cryptography. In this talk, I will describe the structure of isogeny graphs of CM curves, and of oriented supersingular curves, and their relationship to the structure of supersingular isogeny graphs. In particular, the endomorphism ring of a supersingular elliptic curve is an order in a quaternion algebra. Embeddings of imaginary quadratic orders into this quaternion order are called \emph{orientations}. Explicit knowledge of this endomorphism ring leads to well-known pathfinding algorithms. In joint work, we develop classical and quantum algorithms for path-finding under the assumption that \emph{only one} endomorphism from this order is known (equivalently, one orientation). In related work, we demonstrate a bijection between the cycles in a fixed isogeny graph and the cycles in the union of all CM graphs which cover it. As a result, we count the cycles in the isogeny graph in terms of certain class numbers of imaginary quadratic orders.

This is a joint work with Sarah Arpin, Mingjie Chen, Kristin E. Lauter and Renate Scheidler.