# 2017/2018

• 2017-11-07
• 2017-10-24
José Manuel Rodriguez Caballero (Labri)
Kassel and Reutenauer computed the zeta function of the Hilbert scheme of n points on a two-dimensional torus and showed it satisfies several number-theoretical properties via modular forms. Classifying the singularities of this rational function into zeros and poles, we define a word which contains a lot of number-theoretical information about n (the above-mentioned number of points). This nontrivial connection between natural numbers and words can be used to define many classical subsets of natural numbers in terms of rational and context-free languages (e.g. the set of semi-perimeters of Pythagorean triangles, the set of numbers such that any partition into consecutive parts has an odd number of parts). Also, some arithmetical functions can be described in way (e.g. the Erdös-Nicolas function, the number of middle divisors). Finally, this approach provides a new technique to prove number-theoretical results just using relationships among context-free languages.
• 2017-10-17
We review methods for validated arbitrary-precision numerical computation of elliptic functions and their inverses (the complete and incomplete elliptic integrals), as well as the closely related Jacobi theta functions and $\mathrm{SL}_2(\mathbb{Z})$ modular forms. A general strategy consists of two stages: first, using functional equations to reduce the function arguments to a smaller domain; second, evaluation of a suitable truncated series expansion. For elliptic functions and modular forms, one exploits periodicity and modular transformations for argument reduction, after which the rapidly convergent series expansions of Jacobi theta functions can be employed. For elliptic integrals, a comprehensive strategy pioneered by B. Carlson consists of using symmetric forms to unify and simplify both the argument reduction formulas and the series expansions (which involve multivariate hypergeometric functions). Among other aspects, we discuss error bounds as well as strategies for argument reduction and series evaluation that reduce the computational complexity. The functions have been implemented in arbitrary-precision complex interval arithmetic as part of the Arb library.
• 2017-10-10
• 2017-10-03
• 2017-09-26
• 2017-09-19
• 2017-09-08
Polynômes modulaires
Nous présentons une méthode de calcul de polynômes modulaires par évaluation/interpolation développée pour les courbes elliptiques par Enge et généralisée au genre 2 par Dupont puis par Milio et Robert.

En particulier nous donnons des exemples de polynômes modulaires paramétrisant des isogénies totalement isotrope entre surfaces abéliennes sur l’espace de module de Siegel puis des polynômes concernant les isogénies cycliques entre surfaces abéliennes sur l’espace de module de Hilbert.

• 2017-09-08
Abdoul Aziz Ciss (Université Cheikh Anta Diop, Dakar, Sénégal)
• 2017-09-08
Tony Ezome (Université des Sciences et Techniques de Masuku (USTM), Franceville, Gabon)
Bases normales et variétés Jacobiennes
Nous présentons une généralisation des travaux de Couveignes et Lercier au cas des variétés jacobiennes.
• 2017-09-07
SIDH II
• 2017-09-07
A Linearly Homomorphic Encryption Scheme based on the Discrete Logarithm Problem
We present the first linearly homomorphic encryption scheme whose security relies on the sole hardness of a discrete logarithm problem. Our approach requires some special features of the underlying group. Therefore, the discrete logarithm assumption holds in the class group of a non maximal order of an imaginary quadratic field. Its algebraic structure and unknown order make 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. Under some conditions, the prime $p$ can be scaled to fit the application needs. A notable difference with previous works is that the security does not depend on the hardness of the factorization of integers, which allows in particular to embed the message space into the whole group $\mathbb{Z}/p\mathbb{Z}$.

Joint work with Fabien Laguillaumie.

• 2017-09-07
Comptage de points
• 2017-09-07
Aurel Page (IMB)
Algèbres de quaternions et courbes elliptiques supersingulières III
• 2017-09-06
Aurel Page (IMB)
Algèbres de quaternions et courbes elliptiques supersingulières II
• 2017-09-06
The theory of complex multiplication can be used to obtain elliptic curves with a number of points known in advance, which has applications from primality proving to cryptography. Complex multiplication provides a link between the analytic theory of elliptic curves over the complex numbers, the arithmetic of elliptic curves and explicit class field theory of imaginary-quadratic fields. We give an elementary introduction to this beautiful theory and discuss asymptotically optimal algorithms to compute the needed data, which are used successfully in practice.
• 2017-09-06
Tony Ezome (Université des Sciences et Techniques de Masuku (USTM), Franceville, Gabon)
Bases normales et courbes elliptiques
Nous présentons la construction des bases normales utilisant les courbes elliptiques proposée par Couveignes et Lercier.
• 2017-09-06
• 2017-09-05
Abdoulaye Maiga (Université Cheikh Anta Diop, Dakar, Sénégal)
Lifts canoniques
• 2017-09-05
SIDH I
Nous présentons le cryptosystème post-quantique Supersingular Isogenies Diffie-Hellman proposé par De Feo, Jao et Plut basé sur les graphes d’isogénies des courbes elliptiques supersingulières.
• 2017-09-05
Emmanuel Fouotsa (The University of Bamenda, Cameroon)
Pairings are bilinear maps defined over groups of points of an elliptic curves. They enables many applications in cryptography and the Miller algorithm is the main tool for their computation. In this talk, we explain an alternative way of computing these maps based on the work of K. Stange and then study some recent optimizations. (References.)
• 2017-09-04
Aurel Page (IMB)
• 2017-09-04
• 2017-09-04
Emmanuel Fouotsa (The University of Bamenda, Cameroon)
In this talk, we explain the construction of isogenies graphs and isogenies cycles for ordinary elliptic curves. We illustrate concepts with many examples. We end with a brief presentation and a security analysis of the Rostovtev-Stolbunov cryptosystem. (References.)
• 2017-09-04
Mohamadou Sall (Université Cheikh Anta Diop, Dakar, Sénégal)
A finite Galois extension $E$ of a field $F$ can always be viewed as a vector space over $F$. This gives many possibilities for representing elements of $E$. Normal bases are one of the most useful representations. These concern both mathematical theory and practical applications. In this talk we give a brief overview of normal bases, namely by describing an efficient arithmetic over finite field.