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Mathematics of Computation

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Computing $ (\ell,\ell)$-isogenies in polynomial time on Jacobians of genus $ 2$ curves


Authors: Romain Cosset and Damien Robert
Journal: Math. Comp. 84 (2015), 1953-1975
MSC (2010): Primary 11Y40, 14K02; Secondary 94A60, 14G50, 11T71
DOI: https://doi.org/10.1090/S0025-5718-2014-02899-8
Published electronically: November 18, 2014
MathSciNet review: 3335899
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Abstract: In this paper, we compute $ \ell $-isogenies between abelian varieties over a field of characteristic different from $ 2$ in polynomial time in $ \ell $, when $ \ell $ is an odd prime which is coprime to the characteristic. We use level $ n$ symmetric theta structure where $ n=2$ or $ n=4$. In the second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus $ 2$ hyperelliptic curves to theta coordinates of level $ 2$ or $ 4$. Combined with the preceding algorithm, this gives a method to compute $ (\ell ,\ell )$-isogenies in polynomial time on Jacobians of genus $ 2$ curves.


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Additional Information

Romain Cosset
Affiliation: Campus Scientifique, Loria, 54506 Vandouevre-Les-Nancy, France
Email: romain.cosset@crans.org

Damien Robert
Affiliation: Universite Bordeaux 1, Institut Mathematiques de Bordeaux, 351 Cours de la Liberation, Batiment A33, 33405 Talence, Cedex France
Email: damien.robert@inria.fr

DOI: https://doi.org/10.1090/S0025-5718-2014-02899-8
Received by editor(s): March 24, 2011
Received by editor(s) in revised form: October 4, 2013
Published electronically: November 18, 2014
Article copyright: © Copyright 2014 American Mathematical Society