The VI AMMCS International Conference

Waterloo, Ontario, Canada | August 14-18, 2023

AMMCS 2023 Plenary Talk

Supersingular isogeny graphs and orientations

Katherine E. Stange (University of Colorado)

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.
Katherine E. Stange is a number theorist and cryptographer at the University of Colorado, Boulder. She is in love with the rich structure and variation of number theory, with its potential to interact with geometry and illustration, and is fascinated by its relationship to human affairs through cryptography. She is happiest simply wandering around taking field notes on the behaviour of mathematical flora and fauna. Over the years, she has happily trailed elliptic curves and isogenies, quadratic forms, Kleinian groups, Apollonian circle packings, and continued fractions, to name a few. In real life, she trails her two children, often on two wheels.

She can be found on the web at