Supersingular Curves With Small Non-integer Endomorphisms
Abstract
We introduce a special class of supersingular curves over , 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 when 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 small-degree isogenies 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 -isogeny graphs.
Cite
@article{arxiv.1910.03180,
title = {Supersingular Curves With Small Non-integer Endomorphisms},
author = {Jonathan Love and Dan Boneh},
journal= {arXiv preprint arXiv:1910.03180},
year = {2020}
}
Comments
25 pages, 2 figures; improved bound for Theorem 1.3 (from an improvement to the proof of Proposition 4.5); new Appendix C on l-isogenies that can't be replaced by short coprime-to-l-isogenies; moved discussion of algorithms to an appendix; new, more direct (local) proofs of Lemmas 4.2 and 5.4; many minor revisions