LUBICZ, David; ROBERT, Damien

doi:10.1016/j.ffa.2016.01.009

Métadonnées

Licence d’utilisation du document

LUBICZ, David
Institut de Recherche Mathématique de Rennes [IRMAR]

ROBERT, Damien
Lithe and fast algorithmic number theory [LFANT]
Institut de Mathématiques de Bordeaux [IMB]
Laboratoire International de Recherche en Informatique et Mathématiques Appliquées [LIRIMA]

Langue

Article de revue

Ce document a été publié dans

Finite Fields and Their Applications. 2016-05, vol. 39, p. 130-158

Elsevier

Résumé en anglais

A 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.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/194287

DOI

http://dx.doi.org/10.1016/j.ffa.2016.01.009

Projet Européen

Algorithmic Number Theory in Computer Science

Project ANR

Espaces de paramètres pour une arithmétique efficace et une évaluation de la sécurité des courbes - ANR-12-BS01-0010
SIM et théorie des couplages pour la sécurité de l'information et des communications - ANR-12-INSE-0014
Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation - ANR-11-LABX-0020

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251