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Arithmetic on Abelian and Kummer Varieties

hal.structure.identifierInstitut de Recherche Mathématique de Rennes [IRMAR]
dc.contributor.authorLUBICZ, David
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
hal.structure.identifierLaboratoire International de Recherche en Informatique et Mathématiques Appliquées [LIRIMA]
dc.contributor.authorROBERT, Damien
dc.date.accessioned2024-04-04T03:17:12Z
dc.date.available2024-04-04T03:17:12Z
dc.date.created2015-09-01
dc.date.issued2016-05
dc.identifier.issn1071-5797
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/194287
dc.description.abstractEnA Kummer variety is the quotient of an abelian variety by the automorphism $(-1)$ acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the $x$-coordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, there remains enough arithmetic to enable the computation of exponentiations via a Montgomery ladder based on differential additions. In this paper, we explain that the arithmetic of a Kummer variety is much richer than usually thought. We describe a set of composition laws which exhaust this arithmetic and show that these laws may turn out to be useful in order to improve certain algorithms. We explain how to compute efficiently these laws in the model of Kummer varieties provided by level $2$ theta functions. We also explain how to recover the full group law of the abelian variety with a representation almost as compact and in many cases as efficient as the level $2$ theta functions model of Kummer varieties.
dc.description.sponsorshipEspaces de paramètres pour une arithmétique efficace et une évaluation de la sécurité des courbes - ANR-12-BS01-0010
dc.description.sponsorshipSIM et théorie des couplages pour la sécurité de l'information et des communications - ANR-12-INSE-0014
dc.description.sponsorshipCentre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020
dc.language.isoen
dc.publisherElsevier
dc.title.enArithmetic on Abelian and Kummer Varieties
dc.typeArticle de revue
dc.identifier.doi10.1016/j.ffa.2016.01.009
dc.subject.halInformatique [cs]/Calcul formel [cs.SC]
dc.description.sponsorshipEuropeAlgorithmic Number Theory in Computer Science
bordeaux.journalFinite Fields and Their Applications
bordeaux.page130-158
bordeaux.volume39
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01057467
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01057467v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Finite%20Fields%20and%20Their%20Applications&rft.date=2016-05&rft.volume=39&rft.spage=130-158&rft.epage=130-158&rft.eissn=1071-5797&rft.issn=1071-5797&rft.au=LUBICZ,%20David&ROBERT,%20Damien&rft.genre=article


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