Higher-degree supersingular group actions

Abstract

We investigate the isogeny graphs of supersingular elliptic curves over GF(p²) 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 GF(p), and there is an action of the ideal class group of Q(√︁-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.