Computing isogenies between Abelian Varieties

2 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
3 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : 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.
keyword :
Type de document :
Article dans une revue
Compositio Mathematica, Foundation Compositio Mathematica, 2012, 148 (05), pp.1483--1515. 〈10.1112/S0010437X12000243〉
Domaine :

Littérature citée [19 références]

https://hal.archives-ouvertes.fr/hal-00446062
Contributeur : Damien Robert <>
Soumis le : vendredi 21 septembre 2012 - 19:15:22
Dernière modification le : samedi 23 septembre 2017 - 01:08:11
Document(s) archivé(s) le : vendredi 16 décembre 2016 - 16:41:49

Fichier

isogenies_web.pdf
Fichiers produits par l'(les) auteur(s)

Citation

David Lubicz, Damien Robert. Computing isogenies between Abelian Varieties. Compositio Mathematica, Foundation Compositio Mathematica, 2012, 148 (05), pp.1483--1515. 〈10.1112/S0010437X12000243〉. 〈hal-00446062v3〉

Consultations de
la notice

450

Téléchargements du document