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# Séminaire de Théorie Algorithmique des Nombres

Responsables : Xavier Caruso et Aurel Page

Page du séminaire

• Le 27 janvier 2015 à 10:00
• Salle 385
Andreas Enge (imb)
Class polynomials for abelian surfaces

The complex multiplication method is well-known for elliptic curves, whereit may be used to construct curves used in primality proofs or to implementcrytosystems, in particular pairing-based ones. A similar approach ispossible for abelian surfaces, that are Jacobians of genus 2 curves,with considerable number theoretic complications. I describe an algorithmusing complex floating point approximations with an asymptotically optimalrunning time, that is, quasi-linear in the size of the class polynomialsproduced as output. Our implementation has been used to carry outparallelised record computations and I present experimental data.

(joint work with Emmanuel Thomé)

• Le 27 janvier 2015 à 10:00
• Salle 385
Andreas Enge (imb)
Class polynomials for abelian surfaces

The complex multiplication method is well-known for elliptic curves, whereit may be used to construct curves used in primality proofs or to implementcrytosystems, in particular pairing-based ones. A similar approach ispossible for abelian surfaces, that are Jacobians of genus 2 curves,with considerable number theoretic complications. I describe an algorithmusing complex floating point approximations with an asymptotically optimalrunning time, that is, quasi-linear in the size of the class polynomialsproduced as output. Our implementation has been used to carry outparallelised record computations and I present experimental data.

(joint work with Emmanuel Thomé)

• Le 3 février 2015 à 10:30
• 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-keycryptography. If $$\G$$ is an abelian group (written additively), thenthe Diffie?Hellman protocol in $$\G$$ is composed of four computationsin the form$P \longmapsto [m]P = \underbrace{P + \cdots + P}_{m \text{ times}}$for various points $$P$$ and integers $$m$$; optimising thisscalar multiplication operation is therefore crucial.

In practice, the most efficient contemporary Diffie?Hellmanimplementations are based on elliptic curves, or Jacobians of genus 2curves. But in these groups, computing $$-P$$ is extremely efficient,so we can use the fact that $$[m]\left(\pm P\right) = \pm([m]P)$$ to simplify andspeed up the protocol, identifying $$P$$ with $$-P$$ (formally, we areworking in the quotient set $$\G/\langle\pm1\rangle$$).These compact?? systems offer significant savings in both space(which translates into slightly shorter keys) and computing time(through simpler pseudo-group law formulae).In the elliptic curve context, this amounts to using only$$x$$-coordinates of points and Montgomery?s pseudo-group law.Bernstein?s Curve25519 software, which has become a de facto referenceimplementation of Diffie?Hellman at the 128-bit security level,is a practical example of these techniques in practice.The genus 2 analogue is Kummer surface arithmetic, where we can useparticularly efficient formulae developed by the Chudnovskys, andpopularized in cryptography by Gaudry.

Recent years have seen renewed interest in theGallant?Lambert?Vanstone (GLV) technique for computing $$[m]P$$.Here, we suppose our elliptic curve (or our genus 2 Jacobian) has anefficiently computable non-integer endomorphism $$\phi$$, which whenapplied to elements of $$\G$$ acts like $$[\lambda]$$ (for some largeeigenvalue $$\lambda$$).Suppose we want to compute $$[m]P$$: first we use the Euclideanalgorithm to compute much smaller integers $$a$$ and $$b$$ such that$$a + b\lambda \equiv m \pmod{\#\G}$$, and then we compute$[m]P = [a]P + [b]\phi(P) \ .$The running time of the multiexponentiation depends on the size of $$a$$and $$b$$, while traditional scalar multiplication depends on the sizeof $$m$$. In practice, $$a$$ and $$b$$ have half the bitlength of$$m$$, which means that GLV and its variants can offer us a significantspeedup.

In this talk, we will discuss the adaptation of GLV techniques to$$x$$-coordinate-only and Kummer surface systems. On the practicalside, we will present some experimental results for a new elliptic-curvebased implementation. On the more theoretical side, we will presentsome new formulae for Kummer surface systems with explicit realmultiplication endomorphisms.

• Le 10 février 2015 à 10:00
• Salle 385
Co-volume of high-rank subgroups of the units of a number field
Since Zimmert's work in the early 1980's the co-volume (essentially a regulator) of units is known to grow exponentially with the unit-rank. At the other end of the rank scale, Lehmer's 1933 conjecture predicts a strong lower bound for the height of a subgroup of rank 1 of the units. Rodriguez-Villegas made a conjecture that interpolates between these two and applies to any subgroup of the units. We will sketch a recent analytic proof of this conjecture in the case of high-rank subgroups.

This is joint work with Ted Chinburg, Ben McReynolds, Matt Stover and James Sundstrom.

• Le 3 mars 2015 à 10:00
• 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 overfinite fields. Since the length of such a code is determined by thenumber of rational points on the curve, it is desirable to use curveswith as many rational points as possible. We investigate a certainclass of Artin-Schreier curves with an unusually large number ofautomorphisms. Their automorphism group contains a large extraspecialsubgroup. Precise knowledge of this subgroup makes it possible tocompute the zeta functions of these curves. As a consequence, we obtainnew examples of curves that attain the provably maximal (or minimal)number of points over an appropriate field of definition.

This is joint work with Irene Bouw, Wei Ho, Beth Malmskog, PadmavathiSrinivasan and Christelle Vincent.

• Le 10 mars 2015 à 10:00
• Salle 385
Guilhem Castagnos (imb)
Linearly Homomorphic Encryption from DDH
In this talk, we will design a linearly homomorphic encryption schemewhose security relies on the hardness of the decisional Diffie-Hellman(DDH) problem. Our approach requires some special features of theunderlying group. In particular, its order is unknown and it contains asubgroup in which the discrete logarithm problem is tractable.Therefore, our instantiation holds in the class group of a non maximalorder of an imaginary quadratic field. Its algebraic structure makes itpossible to obtain such a linearly homomorphic scheme whose messagespace is the whole set of integers modulo a prime p and which supportsan unbounded number of additions modulo p from the ciphertexts. Anotable difference with previous works is that, for the first time, thesecurity does not depend on the hardness of the factorization ofintegers. As a consequence, under some conditions, the prime p can bescaled to fit the application needs.

Joint work with Fabien Laguillaumie.

• Le 31 mars 2015 à 10:00
• Salle 385
Karim Belabas (imb)
Modular symbols and p-adic L functions I

• Le 7 avril 2015 à 10:00
• Salle 385
Karim Belabas (imb)
Modular symbols and p-adic L functions II

• Le 14 avril 2015 à 10:00
• Salle 385
Karim Belabas (imb)
Modular symbols and p-adic L functions III

• Le 5 mai 2015 à 10:00
• 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
• 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
• Salle 385
Iuliana Ciocanea-Teodorescu (Leiden+IMB)
Algorithms for finite rings
We will discuss deterministic polynomial time algorithms designed toanswer a series of fundamental questions about finite rings and finitemodules. These include the module isomorphism problem, computing theminimum 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
• Salle 385
Andreas Enge (imb)
Optimised addition sequences for eta and theta functions
The main ingredient of complex multiplication algorithms for ellipticcurves that compute class and modular polynomials via floating pointapproximations is the evaluation of Dedekind?s ?- and of more general?-functions. While algorithms are known that are asymptoticallyquasi-linear in the desired precision, in practice it is usually fasterto evaluate lacunary power series. It has been observed experimentallythat particularly short addition sequences exist for the speciallystructured exponents of ? and ?. A leisurely stroll through classicnumber theory will provide us with proofs of this fact.

Joint work in progress with William Hart and Fredrik Johansson.

• Le 23 juin 2015 à 10:00
• Salle 385
Enea Milio (imb)
Multiplication réelle et polynômes modulaires
Soit $K=\mathbb{Q}(\sqrt{2})$ ou $\mathbb{Q}(\sqrt{5})$. Il existe deuxinvariants qu?on appelle invariants de Gundlach qui engendrent le corpsdes fonctions modulaires symétriques de Hilbert. Si $\beta$ est unélément totalement positif de $O_K$ de norme $p$, les $\beta$-polynômesmodulaires paramétrisent les classes d?isomorphisme de variétésabéliennes principalement polarisées ayant multiplication réelle par$O_K$ et munis d?une $\beta$-isogénie ou d?une $\beta^c$-isogénie. Nousdécrivons un algorithme efficace pour calculer ces polynômes entransposant certains calculs sur l?espace de Siegel. Nous étendrons cesméthodes à des invariants dérivés des fonctions thêta.
• Le 8 septembre 2015 à 10:00
• 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 oninteger factorization and discrete logarithm computation in finitefields. First, I will present the ECM algorithm for integerfactorization and a new method to analyze the elliptic curves used inthis algorithm by studying the Galois properties of divisionpolynomials. Then, I will talk about the NFS algorithm for integerfactorization and in particular the polynomial selection step for whichI will propose improvements of existing algorithms. Finally, I willtalk about a common step of the NFS algorithm for integer factorizationand the NFS-DL and FFS algorithms for discrete logarithm computations:the filtering step. I will explain this step thoroughly and present animprovement for which I will study the impact using data from severalcomputations of discrete logarithms and factorizations.
• Le 22 septembre 2015 à 11:00
• 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 basedprotocols and describe some countermeasures. I will particularly showthat the point blinding countermeasure does not provide a completeprotection to Miller?s algorithm which is the main tool for pairings.
• Le 6 octobre 2015 à 11:00
• 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 integresur $K$ et $\ell$ un nombre premier impair different de lacarcteristique de $K$. Notons $W$ l?ensemble des classes d?equivalencelineaire de diviseurs effectifs de degre 1 sur $C$. Nous nousinteressons aux sections globales d?un faisceau de $O_C$-modules sur lajacobienne $J_C$ de C. Plus precisement nous allons construitre unebase 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
• 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 lastyear on computing various transcendental functions in intervalarithmetic. The first notable result is a large (order of magnitude)speed improvement for elementary functions. The second project concernsgeneralized hypergeometric functions (including the incomplete gammafunction, Bessel functions, and others). This is still a work inprogress, and some significant problems remain, particularly the taskof computing useful enclosures when the inputs are large, inexactcomplex numbers. Finally, I have a fairly complete implementation ofthe classical Jacobi theta functions, elliptic functions and modularforms. I will describe an optimization for theta series, following upthe results presented earlier by Andreas Enge (2015-06-02), and discussthe application of computing class polynomials.
• Le 24 novembre 2015 à 11:00
• Salle 385
Julien Keuffer (Morpho)
The SEA algorithm in PARI/GP
The Schoof-Elkies-Atkin (SEA) algorithm is currently the most efficientalgorithm for counting the number of points of an elliptic curve definedover a finite field of large characteristic. The main idea of thisalgorithm is to use the relation between the order of the curve and thetrace of the Frobenius endomorphism and then to compute this trace modulosmall primes. Using the CRT and the Hasse-Weil bound leads to find theexact value of the trace. The implementation of SEA in PARI/GP is basedon Reynald Lercier?s thesis, published in 1997. Many improvements havebeen proposed since. In this talk, I will present two algorithms(respectively published by Gaudry and Morain and by Mihailescu, Morainand Schost) to compute the trace in the so-called Elkies case, theirimplementations in PARI and comparisons I made during my master?sinternship in the French Network and Information Security Agency.
• Le 3 décembre 2015 à 10:00
• 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 abelianvariety and introduce an approach to characterize it through thecharacter theory of compact Lie groups. We illustrate the method withexamples of generic curves of low genus, with Sato-Tate group$\mathrm{USp}(2g)$; special curves which yield proper subgroups, and afamily of Katz giving rise to Galois representations in$\mathrm{SO}(2g+1)$.

This is joint work with Gilles Lachaud and Yih-Dar Shieh.

• Le 15 décembre 2015 à 11:00
• Salle 385
Bill Allombert (imb)
Les aspect combinatoires des fonctions L d'Artin.

• Le 26 janvier 2016 à 11:00
• Salle 1
Présentation de WIMS (WWW Interactive Multipurpose Server)

• Le 9 février 2016 à 11:00
• Salle 385
Павел Соломатин (imb)
L-functions of Genus Two Abelian Coverings of Elliptic Curves over Finite Fields
Initially motivated by the relations between Anabelian Geometry andArtin’s L-functions of the associated Galois-representations, here westudy the list of zeta-functions of genus two abelian coverings ofelliptic curves over finite fields. Our goal is to provide a completedescription of such a list.
• Le 1er mars 2016 à 11:00
• Salle 385
Cyril Bouvier (imb)
Nonlinear polynomial selection for the number field sieve: improving Montgomery's method
The number field sieve is the most efficient known algorithm forfactoring large integers that are free of small prime factors. The goalof the polynomial selection, the first stage of this algorithm, is tocompute a pair of integer polynomials. Montgomery proposed a method forgenerating two nonlinear polynomials which relies on the constructionof small modular geometric progressions. In this talk, I will presenttheoretical and practical improvements to Montgomery’s method thatallow us to generate pairs of a quadratic and a cubic polynomials andpairs of two cubic polynomials for larger integer that was previouslypossible.Joint work with Nicholas Coxon.
• Le 8 mars 2016 à 11:00
• Salle 385
Fabien Pazuki (IMB et Université de Copenhague)
Régulateurs de corps de nombres et de variétés abéliennes et propriété de Northcott.
Soit $A$ une variété abélienne définie sur un corps de nombres $K$. On peutdéfinir un régulateur associé au groupe de Mordell-Weil des pointsrationnels $A(K)$, lequel joue un rôle important dans la forme forte dela conjecture de Birch et Swinnerton-Dyer. Si l’on suppose vraie laconjecture de Lang et Silverman, on montre alors que ce régulateurvérifie la propriété de finitude suivante : il n’y a qu’un nombre fini devariétés abéliennes simples de dimension fixée $g$, définie sur $K$, derang non nul et de régulateur borné. On montre de plus (dans le courantde la preuve) une inégalité inconditionnelle entre la hauteur deFaltings de $A$, les premiers de mauvaise réduction de $A$ et le rang deMordell-Weil de $A$. L’exposé commencera par une introduction présentantun résultat similaire et inconditionnel pour les régulateurs defamilles de corps de nombres.
• Le 15 mars 2016 à 11:00
• Salle 385
Bill Allombert (imb)
Survey on computing isogeny between elliptic curves.
We present methods to compute isogenies between elliptic curves, and weapply them to the computation of the isogenies matrix of an ellipticcurve defined over the rational and to the Schoof Elkies Atkinalgorithm for counting point on elliptic curves defined over a finitefield.
• Le 22 mars 2016 à 11:00
• Salle 385
Alexandre Le Meur (Université de Rennes)
Formules de Thomae généralisées aux cas des extensions galoisiennes résolubles de $\mathbb{P}^1$.
D’un point de vue classique, les formules de Thomae relient desrapports de puissances de theta constantes avec les coordonnées affinesdes points de ramification d’une courbe hyperelliptique. A partir desannées 80, plusieurs auteurs, ayant des préoccupations centrés sur laphysique, ont montré des généralisations de ces formules au cas descourbes superelliptiques. Plus récemment, Shau Zemel et Hershel Farkasont écrit un livre en utilisant des arguments essentiellementalgébriques. D’un point de vue arithmétique, ces courbes correspondentà des extensions galoisiennes cycliques d’un corps de fonctions $k(x)$.Nous montrerons comment généraliser ces formules au cas des extensionsrésolubles de $k(x)$ et quelles obstructions peuvent survenir.
• Le 5 avril 2016 à 11:00
• Salle 385
Benjamin Matschke (IMB)
A database of rational elliptic curves with given bad reduction
In this talk we present a database of rational elliptic curves with good reduction outside certain finite sets of primes, including the set {2, 3, 5, 7, 11}, and all sets whose product is at most 1000.

In fact this is a biproduct of a larger project, in which we construct practical algorithms to solve S-unit, Mordell, cubic Thue, cubic Thue–Mahler, as well as generalized Ramanujan–Nagell equations, and to compute S-integral points on rational elliptic curves with given Mordell–Weil basis. Our algorithms rely on new height bounds, which we obtained using the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura–Taniyama conjecture (without relying on linear forms in logarithms), as well as several improved and new sieves. In addition we used the resulting data to motivate several conjectures and questions, such as Baker’s explicit abc-conjecture, and a new conjecture on the number of S-integral points of rational elliptic curves.

This is joint work with Rafael von Känel.

• Le 10 mai 2016 à 11:00
• Salle 385
Nicolas Mascot (University of Warwick)
Calcul de représentations galoisiennes modulaires / Computing modular Galois representations
Nous verrons comment la représentation galoisienne modulo l associée àune forme modulaire classique peut être calculée efficacement, enl’isolant dans la torsion de la jacobienne d’une courbe modulaire. Cecipermet notamment de calculer les coefficients a(p) de la forme en tempspolynomial en log p, ce qui en fait la méthode la plus efficace connueà ce jour.

We will explain how the mod l Galois representation attached to aclassical newform may be efficiently computed, by isolating it amongthe l-torsion of a modular jacobian. This yields a way of computing thecoefficient a(p) of the form in time polynomial in log p, which makesit the most efficient methodknown as of today.

• Le 17 mai 2016 à 11:00
• Salle 385
Nicolas Mascot (University of Warwick)
Certification de représentations galoisiennes modulaires / Certifying modular Galois representations
Nous verrons comment les calculs de représentations galoisiennesprésentés dans l’exposé précédent peuvent être certifiés, en s’appuyantsur la conjecture de modularité de Serre et des calculs explicites decohomologie des groupes.

We will show how the Galois representation computations presented inlast week’s talk may be certified, thanks to Serre’s modularityconjecture and explicit group cohomology computations.

• Le 7 juin 2016 à 10:00
• Salle 385
Jared Asuncion (IMB)
Tower decomposition of Hilbert class fields

• Le 11 octobre 2016 à 14:00
• Salle 385
Enea Milio (Inria Nancy Grand Est)
Une implantation en genre 2 de 'Computing functions on Jacobians and their quotients' de Jean-Marc Couveignes et Tony Ezome
Cet article explique comment définir et évaluer des fonctions sur desJacobiennes de courbes de genre $g$ et sur des quotients de tellesJacobiennes par des sous-groupes isotropes maximaux de la$\ell$-torsion, pour $\ell>2$ premier. Pour le cas spécifique du genre2, il est bien connu qu’à partir d’une courbe hyperelliptique $C$ etd’un sous-groupe isotrope maximal $V$, le quotient $\mathrm{Jac}(C)/V$est la Jacobienne d’une courbe hyperelliptique $C’$,$(\ell,\ell)$-isogène à $C$. L’application de $C$ vers$\mathrm{Jac}(D)$ peut être décrite avec des fractions rationnelles dedegré en $O(\ell)$. L’article donne une méthode pour calculer $C’$ etces fractions. Pour notre exposé, nous nous proposons d’exposer lecontenu de ce papier et de parler de l’implantation que nous avonsfaite en genre 2.
• Le 18 octobre 2016 à 10:00
• Salle 385
Gregor Seiler (ETH Zurich)
Computing ray class fields of imaginary quadratic fields

• Le 8 novembre 2016 à 10:00
• Salle 385
Aurélien Focqué
Algorithmes BMSS et Lercier Sirvent pour SEA dans PARI

• Le 22 novembre 2016 à 10:00
• Salle 385
Razvan Barbulescu
A brief history of pairings
Pairings are a relatively new cryptographic tool which have been theobject of many arithmetic works. In the last few years some of thepairings have become obsolete because of the progress on the underlyingproblem of discrete logarithm in finite fields. We propose ourselves tomake a list of pairings constructions, to explain their advantages butalso their weaknesses. The sporadic curves are vulnerable to the Logjamattack and have never been a popular choice. The small characteristiccurves allow a very good arithmetic but are the target of aquasi-polynomial algorithm. The pairings where the characteristic has alow Hamming weight, which eliminate the cost of modular reductions,have been the object of special attacks. When the embedding degree iscomposite the one can use the tower field arithmetic but there are alsotower field attacks.
• Le 17 janvier 2017 à 10:00
• Salle 385
Damien Stehlé (ENS Lyon)
Tuple lattice sieving
Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving SVP require space which (heuristically) grows as $2^{0.2075n+o(n)}$, where n is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity. We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as $2^{0.1887n+o(n)}$. The naive algorithm for this triple sieve runs in time $2^{0.5661n+o(n)}$. With appropriate filtering of pairs, we reduce the time complexity to $2^{0.4812n+o(n)}$ while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous tradeoff between the memory-intensive sieving and the asymptotically slower enumeration. Joint work with Shi Bai, Thijs Laarhoven
• Le 14 mars 2017 à 10:00
• Salle 385
Cécile Pierrot (Centrum Wiskunde & Informatica, Amsterdam)
Nearly sparse linear algebra
Linear algebra is a widely used tool both in mathematics and computerscience, and cryptography is no exception to this rule. Yet, itintroduces some particularities, such as dealing with linear systemsthat are often sparse, or, in other words, linear systems inside whicha lot of coefficients are equal to zero. We propose to enlarge thisnotion to nearly sparse matrices, characterized by the concatenationof a sparse matrix and some dense columns, and to design an algorithmto solve this kind of problems. Motivated by discrete logarithmscomputations on medium and high characteristic finite fields, theNearly Sparse Linear Algebra bridges the gap between classical denselinear algebra problems and sparse linear algebra ones, for whichspecific methods have already been established. Our algorithmparticularly applies on one of the three phases of NFS (Number FieldSieve) which precisely consists in finding a non trivial element ofthe kernel of a nearly sparse matrix.

This is a joint work with Antoine Joux.

• Le 23 mai 2017 à 10:00
• Salle 385
Christophe Petit (Oxford)
Post-quantum cryptography from supersingular isogeny problems?
We review existing cryptographic schemes based on the hardness ofcomputing isogenies between supersingular isogenies, and present someattacks against them. In particular, we present new techniques toaccelerate the resolution of isogeny problems when the action of theisogeny on a large torsion subgroup is known, and we discuss the impactof these techniques on the supersingular key exchange protocol ofJao-de Feo.
• Le 30 mai 2017 à 10:00
• Salle 385
Benjamin Wesolowski (EPFL)
Isogeny graphs of ordinary abelian varieties
Fix a prime number $\ell$. Graphs of isogenies of degree a power of$\ell$ are well-understood for elliptic curves, but not forhigher-dimensional abelian varieties. We study the case of absolutelysimple ordinary abelian varieties over a finite field. We analysegraphs of so-called $\mathfrak l$-isogenies, resolving that, inarbitrary dimension, their structure is similar, but not identical, tothe 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 classof isogenies known as $(\ell, \ell)$-isogenies. These results lead tonew, provable algorithms to navigate in isogeny graphs, withconsequences for the CM-method in genus 2, for computing explicitisogenies, and for the random self-reducibility of the discretelogarithm problem in genus 2 cryptography.
• Le 6 juin 2017 à 10:00
• Salle 385
Guilhem Castagnos (imb)
Encryption Switching Protocols Revisited: Switching modulo p
Last year, Couteau, Peters and Pointcheval introduced a new primitivecalled encryption switching protocols, allowing to switch ciphertextsbetween two encryption schemes. If such an ESP is built with twoschemes that are respectively additively and multiplicativelyhomomorphic, it naturally gives rise to a secure 2-party computationprotocol. It is thus perfectly suited for evaluating functions, suchas multivariate polynomials, given as arithmetic circuits. Couteau etal. built an ESP to switch between Elgamal and Paillier encryptionswhich do n ot naturally fit well together. Consequently, they had todesign a clever variant of Elgamal over Z/nZ with a costly shareddecryption. In this talk, we first present a conceptually simplegeneric construction for encryption switching protocols. We then givean efficient instantiation of our generic approach that uses twowell-suited protocols, namely a variant of Elgamal in Z/pZ and theCastagnos-Laguillaumie encryption defined over class groups of quadratic fields which is additively homomorphic over Z/pZ. Among otheradvantages, this allows to perform all computations modulo a prime pinstead of an RSA modulus. Overall, our solution leads to significantreductions in the number of rounds as well as the number of bitsexchanged by the parties during the interactive protocols. We also showhow to extend its security to the malici ous setting.

Joint work with Laurent Imbert and Fabien Laguillaumie.

• Le 13 juin 2017 à 10:00
• Salle 385
Bernhard Schmidt (Nanyang Technological University, Singapore)
The Anti-Field-Descent Method
A circulant Hadamard matrix of order $v$ is a matrix of the form$H=\begin{pmatrix}a_1 & a_2 & \cdots & a_v \a_v & a_1 & \cdots & a_{v-1} \\cdots & \cdots & \cdots &\cdots \a_2 & a_3 & \cdots & a_1 \\end{pmatrix}$with $a_i=\pm 1$ such that any two rows of $H$ are orthogonal withrespect to the standard inner product. It is conjectured that there isno circulant Hadamard matrix of order larger than $4$.

One way to study circulant Hadamard matrices is the so-calledfield-descent method’’. The essential fact behind this method is thatcertain cyclotomic integers necessarily are contained in relativelysmall fields and thus must have relatively small complex modulus. Inthis talk, I will present a method which reveals a complementaryphenomenon: certain cyclotomic integers cannot be contained in relativelysmall fields and thus must have relatively large complex modulus. Thismethod provides new necessary conditions for the existence of circulantHadamard matrices.

This is joint work with K. H. Leung.

• Le 17 octobre 2017 à 10:00
• Salle 385
Fredrik Johansson (imb)
Numerics of classical elliptic functions, elliptic integrals and modular forms
We review methods for validated arbitrary-precision numericalcomputation of elliptic functions and their inverses (the complete andincomplete elliptic integrals), as well as the closely related Jacobitheta functions and $\mathrm{SL}_2(\mathbb{Z})$ modular forms. A general strategy consists of two stages:first, using functional equations to reduce the functionarguments to a smaller domain; second, evaluation of a suitable truncatedseries expansion. For elliptic functions and modular forms, one exploitsperiodicity and modular transformations for argument reduction, afterwhich the rapidly convergent series expansions of Jacobi theta functionscan be employed. For elliptic integrals, a comprehensive strategypioneered by B. Carlson consists of using symmetric forms to unify andsimplify both the argument reduction formulas and the series expansions(which involve multivariate hypergeometric functions). Among otheraspects, we discuss error bounds as well as strategies for argumentreduction and series evaluation that reduce the computational complexity.The functions have been implemented in arbitrary-precision complexinterval arithmetic as part of the Arb library.
• Le 24 octobre 2017 à 10:00
• Salle 385
José Manuel Rodriguez Caballero (Labri)
Context-free languages in Algebraic Geometry and Number Theory.
Kassel and Reutenauer computed the zeta function of the Hilbert schemeof n points on a two-dimensional torus and showed it satisfies severalnumber-theoretical properties via modular forms. Classifying thesingularities of this rational function into zeros and poles, we definea word which contains a lot of number-theoretical information about n(the above-mentioned number of points). This nontrivial connectionbetween natural numbers and words can be used to define many classicalsubsets of natural numbers in terms of rational and context-freelanguages (e.g. the set of semi-perimeters of Pythagorean triangles,the set of numbers such that any partition into consecutive parts hasan odd number of parts). Also, some arithmetical functions can bedescribed in way (e.g. the Erdös-Nicolas function, the number of middledivisors). Finally, this approach provides a new technique to provenumber-theoretical results just using relationships among context-freelanguages.
• Le 14 novembre 2017 à 10:00
• Salle 385
Jean Kieffer (ENS Paris)
Accélération du protocole d'échange de clés de Couveignes-Rostovtsev-Stolbunov
Ce protocole d’échange de clés est fondé sur la théorie de lamultiplication complexe: un ordre dans un corps quadratique imaginaireagit sur un ensemble de courbes elliptiques ordinaires isogènes définiessur un corps fini. Pour instancier le protocole, on est amené à calculerdes isogénies de différents degrés entre ces courbes à l’aide desalgorithmes développés pour le comptage de points. Ce cryptosystème peutêtre accéléré par un bon choix de courbe elliptique initiale, notammentpar la présence de points de torsion rationnels, et l’on présente uneméthode de recherche de telles courbes.
• Le 20 novembre 2017 à 14:00
• Salle 385
Christian Klein
Computational approach to compact Riemann surfaces
A purely numerical approach to compact Riemann surfaces starting fromplane algebraic curves is presented. The critical points of the algebraiccurve are computed via a two-dimensional Newton iteration. The startingvalues for this iteration are obtained from the resultants with respect toboth coordinates of the algebraic curve and a suitable pairing of theirzeros. A set of generators of the fundamental group for the complement ofthese critical points in the complex plane is constructed from circlesaround these points and connecting lines obtained from a minimal spanningtree. The monodromies are computed by solving the de ning equation of thealgebraic curve on collocation points along these contours and byanalytically continuing the roots. The collocation points are chosen tocorrespond to Chebychev collocation points for an ensuing Clenshaw–Curtisintegration of the holomorphic differentials which gives the periods ofthe Riemann surface with spectral accuracy. At the singularities of thealgebraic curve, Puiseux expansions computed by contour integration on thecircles around the singularities are used to identify the holomorphicdifferentials. The Abel map is also computed with the Clenshaw–Curtisalgorithm and contour integrals. As an application of the code, solutionsto the Kadomtsev–Petviashvili equation are computed on non-hyperellipticRiemann surfaces.
• Le 28 novembre 2017 à 10:00
• Salle 385
Frank Vallentin
Coloring the Voronoi tessellation of lattices
We define the chromatic number of a lattice: It is the least number ofcolors one needs to color the interiors of the cells of the Voronoitesselation of a lattice so that no two cells sharing a facet are ofthe same color. We compute the chromatic number of the irreducible rootlattices and for this we apply a generalization of the Hoffman bound.
• Le 9 janvier 2018 à 10:00
• Salle 385
Fredrik Johansson (imb)
Numerical integration in complex interval arithmetic
We present a new implementation of validated arbitrary-precisionnumerical evaluation of definite integrals $\int_a^b f(x) dx$,available in the Arb library. The code uses a version of the Petrasalgorithm, which combines adaptive subdivision with Gauss-Legendre (GL)quadrature, evaluating the integrand on complex intervals surroundingthe path of integration to obtain rigorous error bounds. The first partof the talk discusses the general algorithm and its performance forinteresting families of integrals. The second part, which is based onjoint work with Marc Mezzarobba, discusses the fast computation of GLquadrature nodes with rigorous error bounds. It is well known that GLquadrature achieves a nearly optimal rate of convergence for analyticintegrands with singularities well isolated from the path ofintegration, but due to the cost of generating GL quadrature nodes, themore slowly converging Clenshaw-Curtis and double exponentialquadrature rules have often been favored when an accuracy of severalhundreds or thousands of digits is required. We consider the asymptoticand practical aspects of this problem. An order-of-magnitude speedup isobtained over previous code for computing GL nodes with simultaneoushigh degree and precision, which makes GL quadrature viable even atvery high precision.
• Le 22 janvier 2018 à 10:00
• Salle 2
Philippe Moustrou (IMB)
On the Density of Sets Avoiding Parallelohedron Distance 1
Let $\Vert \cdot \Vert$ be a norm on $\mathbb{R}^n$. We consider theso-called unit distance graph $G$ associated with $\Vert \cdot \Vert$:the vertices of $G$ are the points of $\mathbb{R}^n$, and the edgesconnect the pairs $\{x,y\}$ satisfying $\Vert x-y\Vert=1$. We define$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ as the supremum of thedensities achieved by independent sets of $G$. The number $m_1$ wasintroduced by Larman and Rogers (1972) as a tool to study themeasurable chromatic number $\chi_m(\mathbb{R}^n)$ of $\mathbb{R}^n$for the Euclidean norm.

The best known estimates for $\chi_m(\mathbb{R}^n)$ and$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$ present relations withEuclidean lattices, in particular with the sphere packing problem.

The determination of $m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)$for the Euclidean norm is still a difficult question. We study thisproblem for norms whose unit ball is a convex polytope. More precisely,if the unit ball corresponding with $\Vert \cdot \Vert$ tiles$\mathbb{R}^n$ by translation, for instance if it is the Voronoiregion of a lattice, then it is easy to see that$m_1\left(\mathbb{R}^n,\Vert \cdot \Vert\right)\geq \frac{1}{2^n}$.

C. Bachoc and S. Robins conjectured that equality always holds. We showthat this conjecture is true for $n=2$ and for some families of Voronoiregions of lattices in higher dimensions.

• Le 30 janvier 2018 à 10:00
• Salle 385
Jared Asuncion (IMB)
ECPP in PARI/GP
The elliptic curve primality proving (ECPP) algorithm not only proves(or disproves) the primality of an integer $N$ but also provides, if$N$ is prime, a primality certificate which one can verify quickly. Inthis talk, we recall the steps of ECPP and discuss its implementationin PARI/GP.
• Le 6 mars 2018 à 10:00
• Salle 385
Takashi Fukuda (Nihon University)
Class number calculation for special number fields
I will talk about TC (an interpreter of multiprecision C language whichI developed), Weber’s problem, Coates’ conjecture and an algorithm ofcalculating p-class group of abelian number fields. I also present myproject trying to implement an algorithm mentioned above to PARI/GPduring my stay at IMB.
• Le 20 mars 2018 à 10:00
• Salle 385
Tristan Vaccon (Université de Limoges)
Sur les équations différentielles $p$-adiques à variables séparables
Les trois dernières décennies ont vu le développement de méthodes etalgorithmes $p$-adiques, notamment :
• la factorisation de polynômes rationnels par lemme de Hensel ;
• les algorithmes de comptage de points de Kedlaya et Lauder, reposant surdes résultats avancés de géométrie arithmétique ;
• le calcul d’isogénies entre courbes elliptiques.

Dans toutes ces méthodes et algorithmes, on passe par des calculs surles nombres $p$-adiques, et le problème de la gestion de la précision yest crucial. Avec Xavier Caruso et David Roe, nous avons développé uneméthode, dite de précision différentielle, pour étudier et gérer laprécision $p$-adique.

Dans cet exposé, nous nous intéresserons à l’application de cetteméthode pour l’étude du calcul d’isogénies entre courbes elliptiquesvia la résolution de certaines équations différentielles $p$-adiques àvariables séparables. Il s’agit d’un travail en commun avec PierreLairez de 2016 qui ne traite que du cas $p>2$. Nous présenterons aussidans cet exposé quelques avancées récentes lorsque $p=2$.

• Le 3 avril 2018 à 10:00
• Salle 385
Alex Bartel (Glasgow University)
Cohen-Martinet heuristics revisited
In the early 1990s Henri Cohen and Jacques Martinet offered aprobabilistic model that explains the behaviour of ideal class groupsof number fields in natural families, generalising earlier work byCohen and Lenstra. There is a lot of numerical evidence for thecorrectness of the model, but very few theorems. In joint work withHendrik Lenstra we revisit the Cohen-Martinet heuristics and, amongother things, disprove them in two different ways, but also lendadditional support for the expectation that they are “basically true”.In this talk I will explain one of these disproofs, and will proposepossible corrections.
• Le 24 avril 2018 à 10:00
• Salle 385
Damien Robert (imb)
Huang's proposal for trilinear maps
In a recent paper, Huang proposed atrilinear map involving abelian varieties over finite fields. In this talkwe present his approach.

We will first begin the talk with a review of the standard pairingsconstructions on an abelian variety.

• Le 22 mai 2018 à 10:00
• Salle 385
Jean Kieffer (ENS Paris)
Étude et implémentation de l’algorithme de Schoof–Elkies–Atkin

• Le 12 juin 2018 à 10:00
• Salle 385
Xavier Caruso (Université de Rennes)
Variations autour d'un théorème de Christol
Un célèbre théorème de Christol affirme qu’une série à coefficientsdans $\mathbb{Z}/p\mathbb{Z}$ est algébrique sur$\mathbb{Z}/p\mathbb{Z}(x)$ si et seulement si la suite de sescoefficients est $p$-automatique.

L’objectif de cet exposé sera de raconter de jolies mathématiques enlien de ce théorème. Je commencerai par esquisser deux démonstrations“classiques” de ce résultat, puis montrerai comment les combiner pourobtenir une variante effective du théorème de Christol. Je présenteraiensuite une application de ce résultat à une question de naturealgorithmique.

Si le temps le permet, je discuterai également les liens entre théorèmede Christol et équations différentielles p-adiques et montrerai commentutiliser ce nouvel ingrédient pour accélérer l’algorithme précédent.

(Travail en commun avec A. Bostan, G. Christol et Ph. Dumas.)

• Le 5 juillet 2018 à 10:00
• Salle 385
Jean-François Biasse (University of South Florida)
Fast multiquadratic S-unit computation and application to the calculation of class groups
Let $L=Q(√d_1, … ,√d_n)$ be a real multiquadratic field and S be a setof prime ideals of L that does not contain any divisors of 2. In thispaper, we present a heuristic algorithm for the computation of theS-class group and the S-unit group that runs in time$Poly(log(∆),Size(S)) e^{Õ(√ln d)}$ where $d=max_{i≤n} d_i$ and ∆ is thediscriminant of L. We use this method to compute the ideal class groupof the maximal order $O_L$ of L in time $Poly(log(∆)) e^{Õ(√log d)}$. When$log(d)≤log(log(∆))^c$ for some constant $c < 2$, these methods run inpolynomial time. We implemented our algorithm using Sage 7.5.1.
• Le 18 septembre 2018 à 10:30
• Salle 385
Aurel Page et Pascal Molin
Mini groupe de travail: calcul des caractères de Hecke

• Le 6 novembre 2018 à 10:00
• Salle 385
Elie Eid (Université de Rennes)
Calcul d'isogénies en genre 2
Étant donné une courbe algébrique $C$ de genre $2$ définie sur un corpsfini $K$ de caractéristique impaire et un sous-groupe isotrope maximal$\mathcal{V}$ (pour le couplage de commutateur) de l’ensemble despoints de $l$-torsion où $l$ est un entier (premier) impair, noussavons que le quotient de la jacobienne $J_K(C)$ de $C$ par$\mathcal{V}$ est une variété abélienne de dimension 2 et donc lajacobienne d’une courbe $D$ de genre $2$.

La surjection canonique $\pi_l \: : J_K(C) \longrightarrow J_K(D) =J_K(C) / \mathcal{V}$ est une $(l,l)$-isogénie de noyau $\mathcal{V}$.

Dans cet exposé, on s’intéresse à l’algorithme de Couveignes et Ezomepour trouver l’équation de la courbe $D$ à partir de sa kummerconstruite à l’aide de fonctions définies sur la jacobienne de lacourbe de départ et son quotient, ainsi que les équations quidéfinissent l’isogénie $\pi_l$.

• Le 27 novembre 2018 à 10:00
• Salle 385
Ida Tucker
Practical fully secure unrestricted inner product functional encryption modulo a prime p. (Chiffrement fonctionel sans restrictions pour le calcul de produits scalaires modulo un nombre premier.)
Functional encryption (FE) is an advanced cryptographic primitive whichallows, for a single encrypted message, to finely control how muchinformation on the encrypted data each receiver can recover. To thisend many functional secret keys are derived from a master secret key.Each functional secret key allows, for a ciphertext encrypted under theassociated public key, to recover a specific function of the underlyingplaintext.

However constructions for general FE are far from practical, or rely onnon-standard and ill-understood cryptographic assumptions.

In this talk I will focus on the construction of efficient FE schemesfor linear functions (i.e. the inner product functionality), and theframework in which our constructions hold. Such schemes yield manypractical applications, and our constructions are the first FE schemesfor inner products modulo a prime that are both efficient and recoverthe result whatever its size. Our framework consist of a cyclic group$G$ where the decision Diffie-Hellman assumption holds together with asubgroup $F$ of $G$ where the discrete logarithm problem is easy. Thissetting can be instantiated with class groups of imaginary quadraticfields, and allows us to encode information in the exponent of thesubgroup $F$, which can be efficiently recovered whatever its size.

• Le 4 décembre 2018 à 11:00
• Salle 385
Aurel Page
Torsion analytique et torsion de Reidemeister en théorie des nombres
Je ferai un exposé de style groupe de travail sur le rôle de la torsiondans l’homologie des groupes arithmétiques en théorie des nombres ; jeprésenterai une méthode permettant d’obtenir de l’information dessus : laformule de Cheeger–Mueller, et ses utilisations notamment parBergeron–Venkatesh et Calegari–Venkatesh. Je parlerai aussi d’untravail que je viens de commencer avec Michael Lipnowski et JeanRaimbault, dont les aspects algorithmiques ont des points communs avecles méthodes de calcul de valeurs de fonctions L.
• Le 11 décembre 2018 à 10:00
• Salle 385
Aurel Page
Torsion analytique et torsion de Reidemeister en théorie des nombres 2

• Le 18 décembre 2018 à 10:00
• Salle 385
Bill Allombert (imb)
Sur le calcul de automorphismes d'un extension Galoisienne niltpotente de corps de nombres.
Je présente un nouvel algorithme en temps polynomial sous GRH pour lecalcul des automorphismes d’une extension Galoisienne de corps denombres sous la condition que le groupe de Galois soit nilpoltent. Cetalgorithme est basé sur la présentation des groupes nilpoltents et lerelèvement des automorphismes de Frobenius et évite la couteusereconnaissance de nombres algébriques par réduction de réseau tout enévitant le cout exponentiel des méthodes combinatoires utilisées dansma thèse.
• Le 19 février 2019 à 10:00
• Salle 385
David Lubicz
Improving the AGM point counting algorithm

• Le 9 avril 2019 à 11:30
• Salle 385
Xavier Caruso (imb)
Vers les codes de Gabidulin géométriques
Dans cet exposé, je commencerai par rappeler la définition et lesprincipales propriétés de codes de Reed-Solomon. Je présenterai ensuitedeux extensions classiques de ces codes, à savoir, d’une part, lescodes géométriques et, d’autre part, les codes de Gabidulin. Ces deuxextensions appaissent toutefois comme orthogonales : du point de vuepratique, elles gomment des limitations différentes de codes deReed-Solomon tandis que, du point de vue technique, elles son basées surdes constructions mathématiques également très différentes. Dans unedeuxième partie de l’exposé, je présenterai quelques idées et quelquesrésultats en vue d’une généralisation commune des codes géométriques etdes codes de Reed-Solomon.
• Le 28 mai 2019 à 10:00
• Salle 385
Francesco Battistoni (University of Milan)
A conjectural improvement for inequalities involving the regulator of number fields
Given the family of number fields with fixed signature, there existsonly a finite number of such fields having regulator less than aprescribed bound: this is due to a classical inequality by Remak,generalized years later by Friedman, which bounds the discriminant of anumber field by means of some terms which depend also on the regulator.

Between 2016 and 2018, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposogave a complete classification of number fields with low regulator for anydegree $\leq 7$ and for totally real and totally complex octic fields,relying both on Remak-Friedman’s inequalities and on a procedurederived by an explicit formula for the Dedekind Zeta function.

In joint work with Giuseppe Molteni, we propose a conjectural improvementof the upper bounds for the discriminant which would allow, using thesame method, to give a classificaton for other signatures in degree 8 and9 : the main conjecture deals with the sharpest estimate for a factorwhich in fact depends on the signature of the fields.

• Le 4 juin 2019 à 10:00
• Salle 385
Corentin Darreye (imb)
Équirépartition de sommes de coefficients de formes modulaires en progression arithmétique.
Après avoir rappelé des résultats classiques d’équirépartition desommes d’exponentielles, j’expliquerai en quoi ce genre de propriétéspermet de mieux comprendre les sommes de coefficients de Fourier deformes modulaires en progression arithmétique. Je donnerai un aperçu dece qui a été démontré auparavant dans cette thématique pour mieuxintroduire certaines questions restant ouvertes auxquelles jem’intéresse, notamment le cas des formes modulaires de poidsdemi-entier.
• Le 10 septembre 2019 à 09:30
• Salle 2
David Roe (MIT)
The inverse Galois problem for p-adic fields
We describe a method for counting the number of extensions of$\mathbb{Q}_p$ with a given Galois group $G$, founded upon thedescription of the absolute Galois group of $\mathbb{Q}_p$ due toJannsen and Wingberg. Because this description is only known for odd$p$, our results do not apply to $\mathbb{Q}_2$. We report on theresults of counting such extensions for $G$ of order up to $2000$(except those divisible by 512), for $p = 3$, 5, 7, 11, 13. Inparticular, we highlight a relatively short list of minimal $G$ that donot arise as Galois groups. Motivated by this list, we prove two theoremsabout the inverse Galois problem for $\mathbb{Q}_p$: one giving anecessary condition for G to be realizable over $\mathbb{Q}_p$ and theother givinga sufficient condition.
• Le 17 septembre 2019 à 10:00
• Salle 2
Fredrik Johansson (imb)
Fungrim : The Mathematical Functions Grimoire
Fungrim is a new, open source database offormulas and tables for mathematical functions. All formulas are representedin symbolic, computer-readable form and include explicit conditions for thevariables.

The immediate goal is to create a web-based special functions reference workthat addresses some of the drawbacks of resources such as the NIST DigitalLibrary of Mathematical Functions, the Wolfram Functions site, and Wikipedia.A potential longer-term ambition is to provide a software library forsymbolic knowledge about special functions, usable by computer algebrasystems and theorem proving software.

This talk will discuss the motivation behind the project, design issues,and possible applications.

• Le 24 septembre 2019 à 10:00
• Salle 2
Jean Kieffer (imb)
Computing isogenies from modular equations in genus 2
Given two elliptic curves such an isogeny of degree l exists between them,there is an algorithm, due to Elkies, that uses modular equations tocompute this isogeny explicitly. It is an essential tool in the SEA pointcounting algorithm: using isogenies is superior to Schoof’s original ideaof using endomorphisms. In this work, we present the analogue of Elkies’algorithm for Jacobians of genus 2 curves, thus opening the way to usingisogenies in higher genus point counting.
• Le 1er octobre 2019 à 10:00
• Salle 2
Damien Robert (imb)
An overview of isogeny algorithms
Let $A$ be an abelian variety and $K$ a finite subgroup. We will discussseveral approaches to compute the isogeny $A \mapsto A/K$, starting fromVélu’s algorithm for elliptic curves, and then the isogeny theorem for thetafunctions, Couveignes and Ezome’s work on Jacobians of curves, and recentprogress with David Lubicz.
• Le 8 octobre 2019 à 10:00
• Salle 2
Jared Asuncion (imb)
Computing Hilbert class fields of quartic CM fields using Complex Multiplication
The Hilbert class field $H_K(1)$ is the maximal unramified abelianextension of $K$. For imaginary quadratic number fields $K$, it can begenerated using special values of certain analytic, modular functions.For quartic CM-fields $K$, the corresponding construction yields only asubfield of $H_K(1)$.

Ray class fields are generalizations of Hilbert class fields. For apositive integer $m > 0$, the ray class field $H_K(m)$ is obtained byrelaxing the ramification conditions for ideals of $\mathcal{O}_K$dividing $m$.

It turns out that there is a particular subfield $L(m)$ of $H_K(m)$which can be generated using special values of higher-level modularfunctions and Stark’s conjectures. For some values of $m$, this $L(m)$contains the Hilbert class field $H_K(1)$. Thus, we can compute theHilbert class field as a subfield of $L(m)$. In this talk, we find anupper bound for such an integer $m$.

If time permits, we will discuss how to compute the Hilbert class fieldas a subfield of this $L(m)$ when $m = 2$.

• Le 15 octobre 2019 à 10:00
• Salle 2
Gilles Zémor
Cryptographie post-quantique à base de codes
Nous nous proposons de faire un état de l’art et de discuter l’état actuelde la cryptologie basée sur les codes.Nous nous intéresserons à l’approche historique, le paradigme de McEliece,ainsi qu’à la méthodologie plus moderne, initiée par Alekhnovich, et inspirée dela cryptologie basée sur les réseaux suite aux travaux d’Ajtai et de Regev enparticulier. Cette deuxième approche ne prétendait pas à l’origine déboucher surdes systèmes de chiffrement compétitifs, mais présentait l’avantage théoriqued’avoir des preuves de sécurité bien identifiées et reconnues par la communauté de complexitéalgorithmique et de cryptologie théorique. Nous détaillerons les principes deces preuves de sécurité qui ne sont pas accessibles de manière évidente dans lalittérature. Nous montrerons également en quoi il y a aujourd’hui convergencedes deux approches du chiffrement basé sur les codes.

Nous parcourrons et ferons une synthèse des propositions actuelles à lacompétition du NIST. Nous nous intéresserons également aux primitives de signature àbase de codes, domaine sensiblement moins développé que le chiffrement.

• Le 22 octobre 2019 à 10:00
• Salle 2
Développeurs LFANT (IMB)
Hacking session

• Le 29 octobre 2019 à 10:00
• Salle 2
Développeurs LFANT (IMB)
Hacking session

• Le 5 novembre 2019 à 10:00
• Salle 2
Henri Cohen (imb)
Apéry-Like recursions and modular forms
Following Zagier and Beukers, we show that the sequencesused by Apery in his proofs of the irrationality ofzeta(2) and zeta(3) are special cases of more general sequenceshaving surprisingly only integer values, and that manyof these sequences can be parametrized by modular forms.Following Almkwist and Zudilin, we also explain that the degreethree sequences used for zeta(3) and generalizations can beautomatically obtained via a Clausen type hypergeometric identityfrom the degree two sequences used for zeta(2) and generalizations.
• Le 8 novembre 2019 à 14:00
• Salle de conférences
Guilhem Castagnos (imb)
HDR defense: Cryptographie basée sur les corps quadratiques: cryptanalyse, primitives et protocoles

• Le 19 novembre 2019 à 10:00
• Salle 2
Maria Dostert (EPFL)
Exact Semidefinite Programming Bounds for Packing Problems
Semidefinite Programming (SDP) is a powerful tool to obtainupper bounds for packing problems. For example, one can consider thekissing problem of the hemisphere in dimension 8 which asks for themaximal number of pairwise non-overlapping spheres which cansimultaneously touch a central hemisphere in 8-dimensional Euclideanspace. The E8 lattice gives a kissing configuration of 183 points.Moreover, using an SDP given by Bachoc and Vallentin one gets an upperbound of 182.99999999996523. Hence, the optimal value is 183. But howcan we obtain the exact rational solution of the SDP based on thefloating point results given by the SDP solver?

In this talk, I will explain how we can determine the exact result ofthe SDP. Furthermore, we use these techniques to obtain exact resultsfor the kissing problem of the hemisphere in dimension 8 as well asfor other packing problems.

Using the exact rational solution for the kissing problem of thehemisphere, we can prove that the optimal kissing configuration isunique up to isometry.

This is a joint work with David de Laat and Philippe Moustrou.

• Le 26 novembre 2019 à 10:00
• Salle 1
Alice Pellet-Mary (ÉNS de Lyon)
An LLL Algorithm for Module Lattices
A lattice is a discrete subgroup (i.e., $\mathbb Z$-module) of $\mathbb R^n$(where $\mathbb Z$ and $\mathbb R$ are the sets of integers and real numbers). The LLL algorithm is a central algorithm to manipulate lattice bases. It takesas input a basis of a Euclidean lattice, and, within a polynomialnumber of operations, it outputs another basis of the same lattice butconsisting of rather short vectors.

On the cryptographic side, many algorithms based on lattices in factuse structured lattices, in order to improve the efficiency of theschemes. Most of the time, these structured lattices are $R$-modules ofsmall dimension, where $R$ is the ring a integers of some number field.It is then tempting to try and adapt the LLL algorithm, which worksover lattices (i.e., $\mathbb Z$-modules), to these $R$-modules.

All the previous works trying to solve this question focused on ringsof integers $R$ that were Euclidean, as the LLL algorithm over $\mathbb Z$crucially rely on the Euclidean division. In this talk, I willdescribe the first LLL algorithm which works in any ring of integers$R$. This algorithm is heuristic and runs in quantum polynomial time ifgiven access to an oracle solving the closest vector problem in afixed lattice, depending only on the ring of integers R.

This is a joint work with Changmin Lee, Damien Stehlé and Alexandre Wallet

• Le 10 décembre 2019 à 10:00
• Salle 2
Développeurs LFANT (IMB)
Hacking session

• Le 10 décembre 2019 à 10:00
• Salle 2
Développeurs LFANT (IMB)
Hacking session

• Le 14 janvier 2020 à 10:00
• Salle 385
Abdoulaye Maiga (IMB)
Canonical Lift of Genus 2 Curves
Let $\mathcal{A}/\mathbb{F}_q$ (with $q=p^n$) be an ordinary abelian variety,a classical result due to Lubin, Serre and Tate says that there exists aunique abelian variety $\tilde{\mathcal{A}}$ over $\mathbb{Z}_q$ such that themodulo $p$ reduction of $\tilde{\mathcal{A}}$ is $\mathcal{A}$ and $End(\tilde{\mathcal{A}})\cong End(\mathcal{A})$ as a ring. In 2000 T.Satohintroduced a point-counting algorithm on elliptic curves over $\mathbb{F}_q$based on canonical lift. In fact the action of the lifted Verschiebung on thetangent space gives Frobenius eigenvalues and hence the characteristicpolynomial of the ordinary elliptic curves over $\mathbb{F}_q$.We propose to extend the canonical lift algorithm introduced by T.Satoh togenus 2 curves over finite fields, using the modular polynomials in dimension 2. We first prove the Kronecker condition in dimension 2 case andthen succeed to lift the endomorphism ring of $\mathcal{A}$ in dimension 2case using a general lift algorithm of a $p$-torsion group of an ordinaryabelian variety. These results provide an algorithm to compute thecharacteristic polynomial of a genus 2 curves in quasi-quadratic timecomplexity.
• Le 28 janvier 2020 à 10:00
• Salle 385
Jacques Martinet (IMB)
Réseaux, variétés abéliennes et courbes
On expliquera d’abord comment la notion de variété abélienne complexe polariséepossède une version euclidienne dans laquelle on considère des triplets $(E,\Lambda,v)$ d’un espace euclidien $E$, d’un réseau $\Lambda$ de $E$ et d’un élément $v$ de $\mathrm{GL}(E)$ tel que $v^2=-\mathrm{Id}$ et $v(\Lambda)\subset\Lambda^*$.

On s’intéressera à de telles données compatibles avec l’action d’un groupe $G\subset\mathrm{SO}(E)$, et l’on décrira plus précisément deux situations :

$\bullet$ Action d’un groupe d’ordre 7 en dimension réelle $6$, une sorte d’appendice à l’article de Elkies paru dansThe Eightfold Way (The beauty of Klein’s quartic curve), Cambridge University Press (1999); S. Levy ed.

$\bullet$ Actions de groupes « pas trop petits » en dimension $4$ en relation avec les courbes de genre $2$.

Dans tous les cas l’ingrédient important est le théorème de Torelli.

• Le 4 février 2020 à 10:00
• Salle 385
Aude Le Gluher (LORIA)
Une approche géométrique efficace pour le calcul d'espaces de Riemann-Roch : Algorithme et Complexité

Le calcul effectif de bases d’espaces de Riemann-Roch intervient dans de nombreux domaines pratiques, notamment pour l’arithmétique dans les jacobiennes de courbes ou dans des codes correcteurs d’erreurs algébraico-géométriques. Nous proposons une variante probabiliste de l’algorithme de Brill et Noether décrit par Goppa pour le calcul d’une base de l’espace de Riemann-Roch $L(D)$ associé à un diviseur $D$ d’une courbe projective plane nodale $C$ sur un corps parfait $k$ suffisamment grand.
On prouve que sa complexité (estimée par le nombre d’opérations arithmétiques dans le corps $k$ est en $O\big(\max\big(\deg(C)^{2\omega}, \deg(D^+)^{\omega}\big)\big)$ où $\omega > 2,38$ est la constante de l’algèbre linéaire et $D^+$ le plus petit diviseur effectif vérifiant $D^+ \geq D$. Cet algorithme probabiliste peut échouer mais sous quelques conditions, on prouve que sa probabilité d’échec est bornée par $O\big(\max\big(\deg(C)^4, \deg(D^+)^2\big)/|E|\big)$ où $E$ est un sous ensemble fini de $k$ dans lequel on peut choisir des éléments de $k$ uniformément aléatoirement.
À notre connaissance cette borne sur la complexité est la meilleure obtenue jusqu’alors pour le calcul d’espaces de Riemann-Roch dans un cadre général. Dans le contexte du calcul de la loi de groupe dans la jacobienne d’une courbe lisse, notre borne améliore aussi la meilleure borne connue à ce jour, due à Khuri-Makdisi. Notre algorithme jouit également du fait que son efficacité repose sur deux blocs pour lesquels des algorithmes efficaces existent : algèbre linéaire et arithmétique des polynômes univariés. Nous avons implémenté cet algorithme en C++/NTL. Les résultats expérimentaux obtenus via cette implémentation semblent indiquer une amélioration des temps de calcul par rapport à l’implémentation dans le logiciel de calcul formel Magma (jusqu’à 200 fois plus rapide sur certaines instances sur de grands corps finis).

• Le 11 février 2020 à 10:00
• Salle 385
Raphael Rieu-Helft (Université Paris-Sud)
How to Get an Efficient yet Verified Arbitrary-Precision Integer Library

We present a fully verified arbitrary-precision integer arithmetic library designed using the Why3 program verifier. It is intended as a verified replacement for the mpn layer of the state-of-the-art GNU Multi-Precision library (GMP).
The formal verification is done using a mix of automated provers and user-provided proof annotations. We have verified the GMP algorithms for addition, subtraction, multiplication (schoolbook and Toom-2/2.5), schoolbook division, divide-and-conquer square root and modular exponentiation. The rest of the mpn API is work in progress. The main challenge is to preserve and verify all the GMP algorithmic tricks in order to get good performance.
Our algorithms are implemented as WhyML functions. We use a dedicated memory model to write them in an imperative style very close to the C language. Such functions can then be extracted straightforwardly to efficient C code. For medium-sized integers (less than 1000 bits, or 100,000 for multiplication), the resulting library is performance-competitive with the generic, pure-C configuration of GMP.

• Le 18 février 2020 à 10:00
• Salle 385
Alex Bartel (University of Glasgow)
The ray class group of a 'random' number field

The Cohen–Lenstra–Martinet heuristics are a probabilistic model for the behaviour of class groups of number fields in natural families. In this talk, I will discuss a generalisation to ray class groups. About 5 years ago, Varma determined the average number of 3-torsion elements in the ray class group of K with respect to m, when m is a fixed rational modulus, and K runs through the family of imaginary quadratic or of real quadratic fields. Since then, Bhargava has been challenging the community to come up with a natural probabilistic model that would explain the numbers obtained by Varma, and to predict more general averages in more general families of number fields. As I will explain in my talk, there turns out to be a very simple-minded way of doing so, and also a much more conceptual one, and they both turn out to be equivalent. The more conceptual one involves an object that does not appear to have been treated in the literature before, but that is very natural: the Aralelov ray class group of a number field. This is joint work with Carlo Pagano.

• Le 10 mars 2020 à 10:00
• Salle 385
Florent Jouve (IMB)
Harmonie et disparités dans le théorème de Chebotarev

Étant donné une extension galoisienne de corps de nombres L/K, le théorème de Chebotarev affirme l’équirépartition des éléments de Frobenius, relatifs aux idéaux premiers non ramifiés, dans les classes de conjugaison de Gal(L/K). On présentera une étude portant sur les variations du terme d’erreur dans le théorème de Chebotarev, lorsque L/K parcourt certaines familles d’extensions. On donnera une formule de transfert pour les fonctions classiques de décompte des nombres (ou idéaux) premiers permettant de ramener la situation à celle d’une extension des rationnels. On exposera enfin quelques conséquences à des problèmes de “type Linnik” et à l’analogue du phénomène de biais de Chebyshev dans les corps de nombres. L’exposé porte sur un travail commun avec D. Fiorilli.

• Le 22 septembre 2020 à 10:00
• Online
Nicolas Mascot (Trinity College Dublin)
Modular Galois representations p-adically using Makdisi's moduli-friendly forms

We will present a p-adic method to compute Galois representations attached to modular forms. This method compares very favourably to the better-known complex-analytic approach. The main ingredient is the use of “moduli-friendly” forms introduced by Makdisi, which allow us to evaluate modular forms at p-adic points of modular curves, and thus to compute in the Jacobian of modular curves without writing down any equations nor q-expansions.

• Le 13 octobre 2020 à 10:00
• Online
Christopher Doris (Heilbronn Institute and University of Bristol)
Computing Galois groups over p-adic fields

We give an overview of the history of computing Galois groups over p-adic fields, with some diversions to recent progress over the rational field. We focus on the “resolvent method,” a family of techniques to compute Galois groups, and present a recent algorithm to do this in general over p-adic fields, the first of its kind. This algorithm greatly increases the degree of polynomial that can be routinely handled, and for example has been used to extend existing databases of Galois groups of p-adic fields to include all degree 18, 20 and 22 extensions of the 2-adic field. The implementation and tables of results are available on the speaker’s website.

• Le 3 novembre 2020 à 10:00
• Online
Samuele Anni (Université Aix-Marseille)
Isomorphismes de représentations galoisiennes modulaires et graphes

Dans cet exposé, je vais expliquer comment tester efficacement et effectivement si deux représentations galoisiennes modulaires du groupe absolu de Galois des rationnels sont isomorphes. En particulier, je présenterai de nouvelles bornes optimales sur le nombre de traces à tester. Je discuterai également brièvement des graphes des isomorphismes, des résultats associés sur les algèbres de Hecke et de la construction d’une base de données de représentations.

• Le 10 novembre 2020 à 10:00
• Online
Raphaël Pagès (IMB)
Calcul efficace des polynômes caractéristiques des p-courbures d'un opérateur différentiel à coefficients entiers

Nous présentons un nouvel algorithme permettant de calculer les polynômes caractéristiques des $p$-courbures d’un opérateur différentiel à coefficients entiers pour tout $p$ premier inférieur à un entier $N$ donné, en temps quasi-linéaire, donc quasi-optimal, en $N$. L’algorithme présenté se base sur les travaux de A. Bostan, X. Caruso et E. Schost ramenant le calcul de cet invariant au calcul d’une factorielle de matrices, ainsi que sur la technique de calcul de factorielles développée par E. Costa, R. Gerbicz et D. Harvey.

• Le 17 novembre 2020 à 10:00
• Online
Fredrik Johansson (IMB)
Calcium: computing in exact real and complex fields

Calcium is a C library for real and complex numbers in a form suitable for exact algebraic and symbolic computation. Numbers are represented as elements of fields $mathbb{Q}(a_1,ldots,a_n)$ where the extension numbers $a_k$ may be algebraic or transcendental. The system combines efficient field arithmetic with automatic construction of fields and ideals of algebraic relations, resulting in a practical computational model of $mathbb{R}$ and $mathbb{C}$ in which equality is rigorously decidable for a large class of numbers which includes $overline{mathbb{Q}}$ as a subset.

• Le 24 novembre 2020 à 10:00
• Online
Anne-Edgar Wilke (IMB)
Recouvrements optimaux d'ensembles de Siegel tronqués par des boules euclidiennes

Étant donné un groupe algébrique $G$ agissant sur un espace affine $V$, il arrive que l’ensemble $V(mathbb{Z})/G(mathbb{Z})$ des orbites entières paramètre des objets arithmétiques et soit donc intéressant à énumérer. Une façon de s’y prendre consiste à expliciter un domaine fondamental de l’action de $G(mathbb{Z})$ sur $V(mathbb{R})$ et à y rechercher les points entiers. Pour cela, on peut essayer de recouvrir ce domaine fondamental par une famille de boules euclidiennes de rayon constant dont le cardinal soit du même ordre de grandeur que le nombre de points entiers. Je montrerai comment mettre en œuvre cette stratégie dans le cas simple de l’action à droite de $mathrm{GL}_n$ sur $mathrm{M}_n$, dont les orbites entières paramètrent les sous-modules de $mathbb{Z}^n$, et pour laquelle on dispose de domaines fondamentaux approchés faciles à décrire, à savoir les ensembles de Siegel.

• Le 1er décembre 2020 à 10:00
• Online
Tommy Hofmann (Saarland University)
The conjugacy problem in $mathrm{GL}(n, mathbb{Z})$

We consider the problem of deciding whether two matrices are conjugate. If the coefficient ring is a field, this problem can be easily solved by using the Jordan normal form or the rational canonical form. For more general coefficient rings, the situation becomes increasingly challenging, both from a theoretical and a practical viewpoint. In this talk, we show how the conjugacy problem for integer matrices can be efficiently decided using techniques from group and number theory. This is joint work with Bettina Eick and Eamonn O’Brien.

• Le 8 décembre 2020 à 10:00
• Salle 385
Alexandre Bailleul (ENS Lyon)
Zéros réels de fonctions L d'Artin et biais de Chebyshev dans les corps de nombres

Le biais de Chebyshev est un phénomène qui a été étudié tout d’abord dans le cadre des “courses de nombres premiers” (Rubinstein et Sarnak, 1994) pour expliquer la prédominance apparente des nombres premiers congrus à 3 mod 4 par rapport à ceux qui sont congrus à 1 mod 4. Ces courses de nombres premiers ont été généralisées notamment dans le contexte des corps de nombres par Ng en 2000. Dans des travaux récents, Fiorilli et Jouve ont étudié le biais de Chebyshev dans des familles d’extensions de corps de nombres, et mis en évidence des comportements limites de type “grandes déviations” et “théorème central limite”. Dans le travail présenté, je mets en évidence l’influence considérable qu’ont certains zéros réels de fonctions L d’Artin sur le biais de Chebyshev dans des extensions particulières de corps de nombres.

• Le 15 décembre 2020 à 10:00
• Salle 385
Elie Eid (Université de Rennes)
TBA

TBA

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