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Title:
An Effective Lower Bound on the Number of Orientable Supersingular Elliptic Curves
Authors: Antonin Leroux
Abstract:In this article, we prove a generic lower bound on the number of \mathfrak{O}-orientable supersingular curves over \mathbb{F}_{p^2}, i.e curves that admit an embedding of the quadratic order \mathfrak{O} inside their endomorphism ring. Prior to this work, the only known effective lower-bound is restricted to small discriminants. Our main result targets the case of fundamental discriminants and we derive a generic bound using the expansion properties of the supersingular isogeny graphs. Our work is motivated by isogeny-based cryptography and the increasing number of protocols based on \mathfrak{O}-oriented curves. In particular, our lower bound provides a complexity estimate for the brute-force attack against the new \mathfrak{O}-uber isogeny problem introduced by De Feo, Delpech de Saint Guilhem, Fouotsa, Kutas, Leroux, Petit, Silva and Wesolowski in their recent article on the SETA encryption scheme.
ePrint: https://eprint.iacr.org/2022/357
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