Abstract

We introduce a special class of supersingular curves over ${\mathbb{𝔽}}_{p^2}$, characterized by the existence of noninteger endomorphisms of small degree. We prove a number of properties about this set. Most notably, we can partition this set into subsets such that curves within each subset have small-degree isogenies between them, but curves in distinct subsets have no small-degree isogenies between them. Despite this, we show that isogenies between distinct subsets can heuristically be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that cannot be efficiently found by searching on $\ell$-isogeny graphs.

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Mathematical Subject Classification 2010

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