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.

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