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Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

hal.structure.identifierUniversiteit Leiden = Leiden University
dc.contributor.authorARPIN, Sarah
hal.structure.identifierUniversity of Bristol [Bristol]
dc.contributor.authorCLEMENTS, James
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
hal.structure.identifierAnalyse cryptographique et arithmétique [CANARI]
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
dc.contributor.authorDARTOIS, Pierrick
hal.structure.identifierNorwegian University of Science and Technology [NTNU]
dc.contributor.authorERIKSEN, Jonathan Komada
hal.structure.identifierEötvös Loránd University [ELTE]
hal.structure.identifierUniversity of Birmingham [Birmingham]
dc.contributor.authorKUTAS, Péter
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierUnité de Mathématiques Pures et Appliquées [UMPA-ENSL]
hal.structure.identifierCentre National de la Recherche Scientifique [CNRS]
dc.contributor.authorWESOLOWSKI, Benjamin
dc.date.accessioned2024-04-04T02:33:26Z
dc.date.available2024-04-04T02:33:26Z
dc.date.issued2023
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190478
dc.description.abstractEnOrientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $\mathfrak{O}$-orientable for a fixed imaginary quadratic order $\mathfrak{O}$ provides non-trivial information towards computing an endomorphism corresponding to the $\mathfrak{O}$-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at $p$ and $\infty$. We provide code implementations in Sagemath which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to $O(p)$, even for cryptographically sized $p$.
dc.description.sponsorshipMéthodes pour les variétés abéliennes de petite dimension - ANR-20-CE40-0013
dc.description.sponsorshipPost-quantum padlock for web browser - ANR-22-PETQ-0008
dc.language.isoen
dc.rights.urihttp://creativecommons.org/licenses/by/
dc.subject.enNumber Theory (math.NT)
dc.subject.enFOS: Mathematics
dc.title.enFinding Orientations of Supersingular Elliptic Curves and Quaternion Orders
dc.typeDocument de travail - Pré-publication
dc.identifier.doi10.48550/arXiv.2308.11539
dc.subject.halMathématiques [math]
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-04186188
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-04186188v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.date=2023&rft.au=ARPIN,%20Sarah&CLEMENTS,%20James&DARTOIS,%20Pierrick&ERIKSEN,%20Jonathan%20Komada&KUTAS,%20P%C3%A9ter&rft.genre=preprint


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