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The main hard problem underlying the field of isogeny-based cryptography is that of finding an isogeny between two given supersingular elliptic curves. An associated (and equivalent) problem is that of finding the endomorphism ring of a given elliptic curve. Indeed, endomorphisms of elliptic curves play a crucial role, and appear in many places in the literature: from the special soundness of the SIDH-based Identification Protocol, to the binding property of Sterner's isogeny graph commitment scheme, to that of finding collisions for the CGL hash function.
From a graph theoretic point of view, these endomorphisms correspond to cycles in . We study these cycles in a very general way, and using two different methods: Brandt matrices and ideal counting. Indeed, this study is not limited to the usual isogeny graph. We instead consider the generalized supersingular -isogeny graph, where is a set of primes and isogenies are allowed to have degree in the set .
Cycles in the generalized supersingular -isogeny graph
Thu, Oct. 30 11am (MATH 220)
Gabrielle Scullard (University of Georgia)
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The study of supersingular elliptic curves oriented by a quadratic imaginary order O, and the horizontal isogenies between them (induced by the action of an ideal in the class group), has become increasingly important in isogeny-based cryptography; these are the curves and isogenies which are fundamental to CSIDH and SCALLOP, for example. We relate N-isogenies between supersingular elliptic curves oriented by an order of discriminant D, to solutions of equations involving positive definite binary quadratic forms of discriminant D. In the case that -DN < 2p, we characterize when non-horizontal N-isogenies arise. As an application, when -D < 2p, we classify when an oriented supersingular elliptic curve has multiple orientations by the order O.
Isogenies between oriented supersingular elliptic curves
Cycles in the generalized supersingular -isogeny graph