Higher-degree supersingular group actions
This work addresses cryptographic security and efficiency in isogeny-based systems, but it appears incremental as it builds on existing concepts like CSIDH and known algorithms.
The paper tackles the problem of generalizing isogeny-based cryptography by studying supersingular elliptic curves with a d-isogeny to their Galois conjugate, investigating both constructive applications like CSIDH generalizations and destructive aspects such as the Delfs-Galbraith algorithm.
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.