The day has come.

Enter a thread on isogenies, random walks and automorphism groups.

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I will explain some of the background we used to write these papers with Ben Smith, so I encourage you to go read them. There are some results at the end of the thread

The main objects in isogeny-based cryptography are elliptic curves and isogenies, usually defined over finite fields. And, of course, isogeny graphs.

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You might want to read about elliptic curves

First of all, an isogeny is a nonconstant morphism of elliptic curves fixing the point at infinity.

So it is a group morphism, a morphism of algebraic curves, it is surjective, and it has finite kernel.