Cryptology ePrint Archive: Report 2021/955

Higher-degree supersingular group actions

Mathilde Chenu and Benjamin Smith

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.

Category / Keywords: public-key cryptography / Isogeny-based cryptography, Supersingular elliptic curves, Endomorphisms

Original Publication (in the same form): MathCrypt 2021 - Mathematical Cryptology