Adding Level Structure to Supersingular Elliptic Curve Isogeny Graphs

Abstract: The classical Deuring correspondence provides a roadmap between supersingular elliptic curves and the maximal orders which are isomorphic to their endomorphism rings. Building on this idea, we add the information of a cyclic subgroup of prime order $N$ to supersingular elliptic curves, and prove a generalisation of the Deuring correspondence for these objects. We also study the resulting $\ell$-isogeny graphs supersingular elliptic curve with level-$N$ structure, and the corresponding graphs in the realm of quaternion algebras. The structure of the supersingular elliptic curve ell-isogeny graph underlies the security of a new cryptographic signature protocol, SQISign, which is proposed to be resistant against both classical and quantum attack.

number theory

Audience: researchers in the topic

London number theory seminar

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