Higher-degree supersingular group actions

Abstract

We investigate the isogeny graphs of supersingular elliptic curves over ${\mathbb{F}}_{p^2}$ equipped with a $d$-isogeny to their Galois conjugate. These curves are interesting because they are, in a sense, a generalization of curves defined over ${\mathbb{F}}_p$, and there is an action of the ideal class group of $\mathbb{Q}(\sqrt{-dp})$ on the isogeny graphs. We investigate constructive and destructive aspects of these graphs in isogeny-based cryptography, including generalizations of the CSIDH cryptosystem and the Delfs-Galbraith algorithm.

Mathematical Cryptology, Florida Online Journals, In press