Adding level structure to supersingular elliptic curve isogeny graphs

Abstract: Supersingular elliptic curve isogeny graphs are a promising new direction for post-quantum cryptography. To ensure the security of new isogeny-based cryptographic protocols, we study the structures underlying the hard problems for those protocols. In this paper, we add the information of level structure to supersingular elliptic curves and study these objects with the motivation of isogeny-based cryptography. For each isomorphism class of supersingular elliptic curve, we add a choice of cyclic subgroup of prime order N. Supersingular elliptic curves with level-N structure map to Eichler orders of level N in a quaternion algebra, just as supersingular elliptic curves map to maximal orders in a quaternion algebra. We study this map and the Eichler orders themselves. Fixing a small prime ell, relatively prime to N and the characteristic of the field, we study ell-isogeny graphs of supersingular elliptic curves with level structure, and how they relate to supersingular isogeny graphs used in post-quantum cryptography.

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