17 research outputs found
The minimum and maximum number of rational points on jacobian surfaces over finite fields
We give some bounds on the numbers of rational points on abelian varieties
and jacobians varieties over finite fields. The main result is that we
determine the maximum and minimum number of rational points on jacobians
varieties of dimension 2
Codes from Jacobian surfaces
This paper is concerned with some Algebraic Geometry codes on Jacobians of
genus 2 curves. We derive a lower bound for the minimum distance of these codes
from an upper "Weil type" bound for the number of rational points on
irreducible (possibly singular or non-absolutely irreducible) curves lying on
an abelian surface over a finite field
The characteristic polynomials of abelian varieties of dimensions 3 over finite fields
We describe the set of characteristic polynomials of abelian varieties of
dimension 3 over finite fields
The characteristic polynomials of abelian varieties of dimensions 4 over finite fields
We describe the set of characteristic polynomials of abelian varieties of
dimension 4 over finite fields
On the Number of Rational Points on Prym Varieties over Finite Fields
International audienceWe give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2
Number of points on abelian and Jacobian varieties over finite fields
We give upper and lower bounds on the number of points on abelian varieties
over finite fields, and lower bounds specific to Jacobian varieties. We also
determine exact formulas for the maximum and minimum number of points on
Jacobian surfaces.Comment: 28 page
Sur le nombre de points rationels des variétés abéliennes sur les corps finis
Le polynôme caractéristique d'une variété abélienne sur un corps fini est défini comme étant celui de son endomorphisme de Frobenius. La première partie de cette thèse est consacrée à l'étude des polynômes caractéristiques de variétés abéliennes de petite dimension. Nous décrivons l'ensemble des polynômes intervenant en dimension 3 et 4, le problème analogue pour les courbes elliptiques et surfaces abéliennes ayant été résolu par Deuring, Waterhouse et Rück.Dans la deuxième partie, nous établissons des bornes supérieures et inférieures sur le nombre de points rationnels des variétés abéliennes sur les corps finis. Nous donnons ensuite des bornes inférieures spécifiques aux variétés jacobiennes. Nous déterminons aussi des formules exactes pour les nombres maximum et minimum de points rationnels sur les surfaces jacobiennes.The characteristic polynomial of an abelian variety over a finite field is defined to be the characteristic polynomial of its Frobenius endomorphism. The first part of this thesis is devoted to the study of the characteristic polynomials of abelian varieties of small dimension. We describe the set of polynomials which occur in dimension 3 and 4; the analogous problem for elliptic curves and abelian surfaces has been solved by Deuring, Waterhouse and Rück.In the second part, we give upper and lower bounds on the number of points on abelian varieties over finite fields. Next, we give lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.AIX-MARSEILLE2-Bib.electronique (130559901) / SudocSudocFranceF
Sur le nombre de points rationnels des variétés abéliennes et des Jacobiennes sur les corps finis
Choosing and generating parameters for low level pairing implementation on BN curves
Many hardware and software pairing implementations can be found in the literature and some pairing friendly parameters are given. However, depending on the situation, it could be useful to generate other nice parameters (e.g. resistance to subgroup attacks, larger security levels, database of pairing friendly curves). The main purpose of this paper is to describe explicitly and exhaustively what should be done to generate the best possible parameters and to make the best choices depending on the implementation context (in terms of pairing algorithm, ways to build the tower field, arithmetic, groups involved and their generators, system of coordinates).
We focus on low level implementations, assuming that additions have a significant cost compared to other operations. However, the results obtained are still valid in the case where additions can be neglected. We also explain why the best choice for the polynomials defining the tower field is only depending on the value of the BN parameter modulo small integers like as a nice application of old elementary arithmetic results. Moreover, we use this opportunity to give some new improvements on arithmetic (in a pairing context) in terms of -addition allowing to save around of them depending on the context