Supersingular Curves With Small Non-integer Endomorphisms

10/08/2019

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by   Jonathan Love, et al.

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We introduce a special class of supersingular curves over F_p^2, characterized by the existence of non-integer endomorphisms of small degree. A number of properties of this set is proved. Most notably, we show that this set partitions into subsets in such a way that curves within each subset have small-degree isogenies between them, but curves in distinct subsets have no isogenies of degree O(p^1/4-ε) between them. Despite this, we show that isogenies between these curves can be computed efficiently, giving a technique for computing isogenies between certain prescribed curves that cannot be reasonably connected by searching on ℓ-isogeny graphs.