Field of moduli and field of definition for curves of genus 2

Cardona, Gabriel; Quer, Jordi

doi:10.1142/9789812701640_0006

Cited by 49 publications

(88 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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“…3. Construct a curve C from a set of roots of H i (x) over F p via the Mestre-Cardona-Quer Algorithm [18], [4], and check if the Jacobian of the curve has order N .…”

Section: Introductionmentioning

confidence: 99%

Generating pairing-friendly parameters for the CM construction of genus 2 curves over prime fields

Lauter

Shang

2012

Des. Codes Cryptogr.

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Abstract. We present two contributions in this paper. First, we give a quantitative analysis of the scarcity of pairing-friendly genus 2 curves. This result is an improvement relative to prior work which estimated the density of pairing-friendly genus 2 curves heuristically. Second, we present a method for generating pairing-friendly parameters for which ρ ≈ 8, where ρ is a measure of efficiency in pairing-based cryptography. This method works by solving a system of equations given in terms of coefficients of the Frobenius element. The algorithm is easy to understand and implement.

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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%