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IMB > Recherche > Séminaires

Séminaire de Théorie Algorithmique des Nombres

Responsables : Razvan Barbulescu et Wessel Van Woerden

Page du séminaire

  • Le 27 janvier 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Andreas Enge imb
    Class polynomials for abelian surfaces
    The complex multiplication method is well-known for elliptic curves, where it may be used to construct curves used in primality proofs or to implement crytosystems, in particular pairing-based ones. A similar approach is possible for abelian surfaces, that are Jacobians of genus 2 curves, with considerable number theoretic complications. I describe an algorithm using complex floating point approximations with an asymptotically optimal running time, that is, quasi-linear in the size of the class polynomials produced as output. Our implementation has been used to carry out parallelised record computations and I present experimental data.
  • Le 27 janvier 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Andreas Enge imb
    Class polynomials for abelian surfaces
    The complex multiplication method is well-known for elliptic curves, where it may be used to construct curves used in primality proofs or to implement crytosystems, in particular pairing-based ones. A similar approach is possible for abelian surfaces, that are Jacobians of genus 2 curves, with considerable number theoretic complications. I describe an algorithm using complex floating point approximations with an asymptotically optimal running time, that is, quasi-linear in the size of the class polynomials produced as output. Our implementation has been used to carry out parallelised record computations and I present experimental data.
  • Le 3 février 2015 à 10:30
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Benjamin Smith INRIA & LIX\, École Polytechnique
    Arithmetic Geometry and Key Exchange : Compact Diffie--Hellman with Efficient Endomorphisms
    $\newcommand{\G}{{\mathcal{G}}}$ Diffie--Hellman key exchange is a fundamental primitive in public-key cryptography. If \(\G\) is an abelian group (written additively), then the Diffie--Hellman protocol in \(\G\) is composed of four computations in the form \[ P \longmapsto [m]P = \underbrace{P + \cdots + P}_{m \text{ times}} \] for various points \(P\) and integers \(m\); optimising this *scalar multiplication* operation is therefore crucial.
  • Le 10 février 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Eduardo Friedman Universidad de Chile
    Co-volume of high-rank subgroups of the units of a number field

  • Le 3 mars 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Renate Scheidler University Calgary
    A family of Artin-Schreier curves with many automorphisms
    Algebraic geometry codes are obtained from certain types of curves over finite fields. Since the length of such a code is determined by the number of rational points on the curve, it is desirable to use curves with as many rational points as possible. We investigate a certain class of Artin-Schreier curves with an unusually large number of automorphisms. Their automorphism group contains a large extraspecial subgroup. Precise knowledge of this subgroup makes it possible to compute the zeta functions of these curves. As a consequence, we obtain new examples of curves that attain the provably maximal (or minimal) number of points over an appropriate field of definition.
  • Le 10 mars 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Guilhem Castagnos imb
    Linearly Homomorphic Encryption from DDH
    In this talk, we will design a linearly homomorphic encryption scheme whose security relies on the hardness of the decisional Diffie-Hellman (DDH) problem. Our approach requires some special features of the underlying group. In particular, its order is unknown and it contains a subgroup in which the discrete logarithm problem is tractable. Therefore, our instantiation holds in the class group of a non maximal order of an imaginary quadratic field. Its algebraic structure makes it possible to obtain such a linearly homomorphic scheme whose message space is the whole set of integers modulo a prime p and which supports an unbounded number of additions modulo p from the ciphertexts. A notable difference with previous works is that, for the first time, the security does not depend on the hardness of the factorization of integers. As a consequence, under some conditions, the prime p can be scaled to fit the application needs.
  • Le 31 mars 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Karim Belabas imb
    Modular symbols and p-adic L functions I

  • Le 7 avril 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Karim Belabas imb
    Modular symbols and p-adic L functions II

  • Le 14 avril 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Karim Belabas imb
    Modular symbols and p-adic L functions III

  • Le 5 mai 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Damien Robert imb
    Arithmetic on Abelian and Kummer varieties I
    The first talk will review the arithmetic of different models of elliptic curves and on the Kummer line. We will also review Mumford coordinates for Jacobian of hyperelliptic curves and introduce theta functions for general abelian varieties.
  • Le 12 mai 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Damien Robert imb
    Arithmetic on Abelian and Kummer varieties II
    The second talk will focus on the arithmetic of theta functions of level 2 and 4 and their use for Abelian and Kummer varieties cryptography.
  • Le 26 mai 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Iuliana Ciocanea-Teodorescu Leiden+IMB
    Algorithms for finite rings
    We will discuss deterministic polynomial time algorithms designed to answer a series of fundamental questions about finite rings and finite modules. These include the module isomorphism problem, computing the minimum number of generators of a module and finding a "good" approximation for the Jacobson radical of a finite ring.
  • Le 2 juin 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Andreas Enge imb
    Optimised addition sequences for eta and theta functions
    The main ingredient of complex multiplication algorithms for elliptic curves that compute class and modular polynomials via floating point approximations is the evaluation of Dedekind's η- and of more general ϑ-functions. While algorithms are known that are asymptotically quasi-linear in the desired precision, in practice it is usually faster to evaluate lacunary power series. It has been observed experimentally that particularly short addition sequences exist for the specially structured exponents of η and ϑ. A leisurely stroll through classic number theory will provide us with proofs of this fact.
  • Le 23 juin 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Enea Milio imb
    Multiplication réelle et polynômes modulaires
    Soit $K=\mathbb{Q}(\sqrt{2})$ ou $\mathbb{Q}(\sqrt{5})$. Il existe deux invariants qu'on appelle invariants de Gundlach qui engendrent le corps des fonctions modulaires symétriques de Hilbert. Si $\beta$ est un élément totalement positif de $O_K$ de norme $p$, les $\beta$-polynômes modulaires paramétrisent les classes d'isomorphisme de variétés abéliennes principalement polarisées ayant multiplication réelle par $O_K$ et munis d'une $\beta$-isogénie ou d'une $\beta^c$-isogénie. Nous décrivons un algorithme efficace pour calculer ces polynômes en transposant certains calculs sur l'espace de Siegel. Nous étendrons ces méthodes à des invariants dérivés des fonctions thêta.
  • Le 8 septembre 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Cyril Bouvier
    Algorithms for integer factorization and discrete logarithms computation
    In this talk, I will present some results obtained during my Ph.D on integer factorization and discrete logarithm computation in finite fields. First, I will present the ECM algorithm for integer factorization and a new method to analyze the elliptic curves used in this algorithm by studying the Galois properties of division polynomials. Then, I will talk about the NFS algorithm for integer factorization and in particular the polynomial selection step for which I will propose improvements of existing algorithms. Finally, I will talk about a common step of the NFS algorithm for integer factorization and the NFS-DL and FFS algorithms for discrete logarithm computations: the filtering step. I will explain this step thoroughly and present an improvement for which I will study the impact using data from several computations of discrete logarithms and factorizations.
  • Le 22 septembre 2015 à 11:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Emmanuel Fouotsa École Normale Supérieure de l'Université de Bamenda
    Analysis of the Efficiency of the point blinding countermeasure against fault attack in Miller's algorithm.
    In this talk, I will present fault attacks against pairing based protocols and describe some countermeasures. I will particularly show that the point blinding countermeasure does not provide a complete protection to Miller's algorithm which is the main tool for pairings.
  • Le 6 octobre 2015 à 11:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Tony Ezome Université des Sciences et Techniques de Masuku\, Franceville
    Constructions et evaluations de fonctions sur les varietes jacobiennes et leur quotients.
    Soient $K$ un corps fini, $C$ une courbe projective absolument integre sur $K$ et $\ell$ un nombre premier impair different de la carcteristique de $K$. Notons $W$ l'ensemble des classes d'equivalence lineaire de diviseurs effectifs de degre 1 sur $C$. Nous nous interessons aux sections globales d'un faisceau de $O_C$-modules sur la jacobienne $J_C$ de C. Plus precisement nous allons construitre une base de l'espace des fonctions $f$ sur $J_C$ tels que le diviseur $div(f)+\ell W$ est un diviseur effectif sur $J_C$.
  • Le 13 octobre 2015 à 11:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Fredrik Johansson imb
    Computing transcendental functions with error bounds
    In this talk, I will give an overview of work I've done in the last year on computing various transcendental functions in interval arithmetic. The first notable result is a large (order of magnitude) speed improvement for elementary functions. The second project concerns generalized hypergeometric functions (including the incomplete gamma function, Bessel functions, and others). This is still a work in progress, and some significant problems remain, particularly the task of computing useful enclosures when the inputs are large, inexact complex numbers. Finally, I have a fairly complete implementation of the classical Jacobi theta functions, elliptic functions and modular forms. I will describe an optimization for theta series, following up the results presented earlier by Andreas Enge (2015-06-02), and discuss the application of computing class polynomials.
  • Le 24 novembre 2015 à 11:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Julien Keuffer Morpho
    The SEA algorithm in PARI/GP
    The Schoof-Elkies-Atkin (SEA) algorithm is currently the most efficient algorithm for counting the number of points of an elliptic curve defined over a finite field of large characteristic. The main idea of this algorithm is to use the relation between the order of the curve and the trace of the Frobenius endomorphism and then to compute this trace modulo small primes. Using the CRT and the Hasse-Weil bound leads to find the exact value of the trace. The implementation of SEA in PARI/GP is based on Reynald Lercier's thesis, published in 1997. Many improvements have been proposed since. In this talk, I will present two algorithms (respectively published by Gaudry and Morain and by Mihailescu, Morain and Schost) to compute the trace in the so-called Elkies case, their implementations in PARI and comparisons I made during my master's internship in the French Network and Information Security Agency.
  • Le 3 décembre 2015 à 10:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    David Kohel Université d'Aix-Marseille
    Characterization of Sato-Tate distributions by character theory
    We describe the generalized Sato-Tate group attached to an abelian variety and introduce an approach to characterize it through the character theory of compact Lie groups. We illustrate the method with examples of generic curves of low genus, with Sato-Tate group $\mathrm{USp}(2g)$; special curves which yield proper subgroups, and a family of Katz giving rise to Galois representations in $\mathrm{SO}(2g+1)$.
  • Le 15 décembre 2015 à 11:00
    • Séminaire de Théorie Algorithmique des Nombres
    Salle 385
    Bill Allombert imb
    Les aspect combinatoires des fonctions L d'Artin.

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