Construction de courbes de genre 2 à partir de leurs modules

Mestre, Jean-François

doi:10.1007/978-1-4612-0441-1_21

Cited by 122 publications

(157 citation statements)

References 7 publications

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“…A first application of the "going up" algorithm is in generating (hyperelliptic) curves of genus 2 over finite fields with suitable security parameters via the CM method. The method is based on first computing invariants for the curve (Igusa invariants) and then using a method of Mestre [33] (see also [34]) to reconstruct the equation of the curve. The computation of the invariants is expensive and there are three different ways to compute their minimal polynomials (the Igusa class polynomials):…”

Section: Motivation For a "Going-up" Algorithmmentioning

confidence: 99%

Isogeny graphs of ordinary abelian varieties

2017

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Fix a prime number . Graphs of isogenies of degree a power of are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the "volcanoes" occurring as graphs of isogenies of elliptic curves. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as ( , )-isogenies: those whose kernels are maximal isotropic subgroups of the -torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.

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Section: Motivation For a "Going-up" Algorithmmentioning

confidence: 99%

Isogeny graphs of ordinary abelian varieties

2017

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“…Over prime fields one can either count points in genus 2 [13], or use the complex multiplication (CM) method for genus 2 [29,39] and 3 [39].…”

Section: Introductionmentioning

confidence: 99%

Aspects of Hyperelliptic Curves over Large Prime Fields in Software Implementations

Avanzi

2004

Lecture Notes in Computer Science

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Abstract. We present an implementation of elliptic curves and of hyperelliptic curves of genus 2 and 3 over prime fields. To achieve a fair comparison between the different types of groups, we developed an ad-hoc arithmetic library, designed to remove most of the overheads that penalize implementations of curve-based cryptography over prime fields. These overheads get worse for smaller fields, and thus for larger genera for a fixed group size. We also use techniques for delaying modular reductions to reduce the amount of modular reductions in the formulae for the group operations.The result is that the performance of hyperelliptic curves of genus 2 over prime fields is much closer to the performance of elliptic curves than previously thought. For groups of 192 and 256 bits the difference is about 14% and 15% respectively.

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“…, it is a simple matter the determination of its Igusa-Clebsch invariants: On the other hand, the determination of an hyperelliptic curve with prescribed invariants I 2 = I 2 , I 4 = I 4 , I 6 = I 6 , I 10 = I 10 is a non-trivial problem, solved by [20] and [4]. We will explain here an elementary method to find a symmetric equation with prescribed invariants, which takes profit of the simplicity of the expressions above.…”

Section: Symmetric Invariants For Genus 2 Curvesmentioning

confidence: 99%

“…There are two main directions in the proposed solutions. The first solution is essentially due to Mestre [20], who considered the case of abelian surfaces. Weber [29] generalized his work to Jacobian varieties of hyperelliptic curves of any genus.…”

Section: Introductionmentioning

confidence: 99%