Higher-degree supersingular group actions

07/19/2021

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by   Mathilde Chenu, et al.

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We investigate the isogeny graphs of supersingular elliptic curves over 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 𝔽_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.