LUBICZ, David; ROBERT, Damien

doi:10.1112/S0010437X12000243

Métadonnées

Licence d’utilisation du document

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

ROBERT, Damien
Cryptology, Arithmetic: Hardware and Software [CARAMEL]
Lithe and fast algorithmic number theory [LFANT]
Institut de Mathématiques de Bordeaux [IMB]

Langue

Article de revue

Ce document a été publié dans

Compositio Mathematica. 2012-09-01, vol. 148, n° 05, p. 1483--1515

Foundation Compositio Mathematica

Résumé en anglais

We describe an efficient algorithm for the computation of isogenies between abelian varieties represented in the coordinate system provided by algebraic theta functions. We explain how to compute all the isogenies from an abelian variety whose kernel is isomorphic to a given abstract group. We also describe an analog of Vélu's formulas to compute an isogenis with prescribed kernels. All our algorithms rely in an essential manner on a generalization of the Riemann formulas. In order to improve the efficiency of our algorithms, we introduce a point compression algorithm that represents a point of level $4\ell$ of a $g$ dimensional abelian variety using only $g(g+1)/2\cdot 4^g$ coordinates. We also give formulas to compute the Weil and commutator pairing given input points in theta coordinates. All the algorithms presented in this paper work in general for any abelian variety defined over a field of odd characteristic.< Réduire

URI

https://oskar-bordeaux.fr/handle/20.500.12278/189823

DOI

http://dx.doi.org/10.1112/S0010437X12000243

Projet Européen

Algorithmic Number Theory in Computer Science

Project ANR

Courbes Hyperelliptiques : Isogénies et Comptage - ANR-09-BLAN-0020

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251