# The geometry of efficient arithmetic on elliptic curves

Abstract : The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripling, by considering families of elliptic curves admiting a $2$-torsion or $3$-torsion point.
Type de document :
Chapitre d'ouvrage
Algorithmic Arithmetic, Geometry, and Coding Theory, AMS Contemporary Mathematics, 637, pp.95-110, 2015

Littérature citée [13 références]

https://hal.archives-ouvertes.fr/hal-01257129
Contributeur : David Kohel <>
Soumis le : samedi 16 janvier 2016 - 08:35:43
Dernière modification le : jeudi 18 janvier 2018 - 01:32:41
Document(s) archivé(s) le : dimanche 17 avril 2016 - 10:32:11

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ec-agct.pdf
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• HAL Id : hal-01257129, version 1
• ARXIV : 1601.03665

### Citation

David Kohel. The geometry of efficient arithmetic on elliptic curves. Algorithmic Arithmetic, Geometry, and Coding Theory, AMS Contemporary Mathematics, 637, pp.95-110, 2015. 〈hal-01257129〉

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