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Faster Algorithms for Isogeny Computations over Extensions of Finite Fields

par Cai, Shiping;Chen, Mingjie ;Petit, Christophe
Référence Lecture notes in computer science, 14966 LNCS, page (136-155)
Publication Publié, 2025-01-01

Article révisé par les pairs

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Résumé :

Any isogeny between two supersingular elliptic curves can be defined over Fp2, however, this does not imply that computing such isogenies can be done with field operations in Fp2. In fact, the kernel generators of such isogenies are defined over extension fields of Fp2, generically with extension degree linear to the isogeny degree. Most algorithms related to isogeny computations are only efficient when the extension degree is small. This leads to efficient algorithms used in isogeny-based cryptographic constructions, but also limits their parameter choices at the same time. In this paper, we consider three computational subroutines regarding isogenies, focusing on cases with large extension degrees: computing a basis of ℓ-torsion points, computing the kernel polynomial of an isogeny given a kernel generator, and computing the kernel generator of an isogeny given the corresponding quaternion ideal under the Deuring correspondence. We then apply our algorithms to the constructive Deuring correspondence algorithm from [EPSV23] in the case of a generic prime characteristic, achieving around 30% speedup over [EPSV23].