2019/2020

• 2019-11-08
HDR defense: Cryptographie basée sur les corps quadratiques: cryptanalyse, primitives et protocoles
• 2019-11-05
Following Zagier and Beukers, we show that the sequences used by Apery in his proofs of the irrationality of zeta(2) and zeta(3) are special cases of more general sequences having surprisingly only integer values, and that many of these sequences can be parametrized by modular forms. Following Almkwist and Zudilin, we also explain that the degree three sequences used for zeta(3) and generalizations can be automatically obtained via a Clausen type hypergeometric identity from the degree two sequences used for zeta(2) and generalizations.
• 2019-10-29
Développeurs LFANT (IMB)
Hacking session
• 2019-10-22
Développeurs LFANT (IMB)
Hacking session
• 2019-10-15
Gilles Zémor
Nous nous proposons de faire un état de l’art et de discuter l’état actuel de 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 de la cryptologie basée sur les réseaux suite aux travaux d’Ajtai et de Regev en particulier. Cette deuxième approche ne prétendait pas à l’origine déboucher sur des systèmes de chiffrement compétitifs, mais présentait l’avantage théorique d’avoir des preuves de sécurité bien identifiées et reconnues par la communauté de complexité algorithmique et de cryptologie théorique. Nous détaillerons les principes de ces preuves de sécurité qui ne sont pas accessibles de manière évidente dans la littérature. Nous montrerons également en quoi il y a aujourd’hui convergence des deux approches du chiffrement basé sur les codes.

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

• 2019-10-08
Computing Hilbert class fields of quartic CM fields using Complex Multiplication
The Hilbert class field $H_K(1)$ is the maximal unramified abelian extension of $K$. For imaginary quadratic number fields $K$, it can be generated using special values of certain analytic, modular functions. For quartic CM-fields $K$, the corresponding construction yields only a subfield of $H_K(1)$.

Ray class fields are generalizations of Hilbert class fields. For a positive integer $m > 0$, the ray class field $H_K(m)$ is obtained by relaxing 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 modular functions and Stark’s conjectures. For some values of $m$, this $L(m)$ contains the Hilbert class field $H_K(1)$. Thus, we can compute the Hilbert class field as a subfield of $L(m)$. In this talk, we find an upper bound for such an integer $m$.

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

• 2019-10-01
An overview of isogeny algorithms
Let $A$ be an abelian variety and $K$ a finite subgroup. We will discuss several approaches to compute the isogeny $A \mapsto A/K$, starting from Vélu’s algorithm for elliptic curves, and then the isogeny theorem for theta functions, Couveignes and Ezome’s work on Jacobians of curves, and recent progress with David Lubicz.
• 2019-09-24
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 to compute this isogeny explicitly. It is an essential tool in the SEA point counting algorithm: using isogenies is superior to Schoof’s original idea of using endomorphisms. In this work, we present the analogue of Elkies’ algorithm for Jacobians of genus 2 curves, thus opening the way to using isogenies in higher genus point counting.
• 2019-09-17
Fungrim is a new, open source database of formulas and tables for mathematical functions. All formulas are represented in symbolic, computer-readable form and include explicit conditions for the variables.

The immediate goal is to create a web-based special functions reference work that addresses some of the drawbacks of resources such as the NIST Digital Library of Mathematical Functions, the Wolfram Functions site, and Wikipedia. A potential longer-term ambition is to provide a software library for symbolic knowledge about special functions, usable by computer algebra systems and theorem proving software.

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

• 2019-09-10
David Roe (MIT)
We describe a method for counting the number of extensions of $\mathbb{Q}_p$ with a given Galois group $G$, founded upon the description of the absolute Galois group of $\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this description is only known for odd $p$, our results do not apply to $\mathbb{Q}_2$. We report on the results of counting such extensions for $G$ of order up to $2000$ (except those divisible by 512), for $p = 3$, 5, 7, 11, 13. In particular, we highlight a relatively short list of minimal $G$ that do not arise as Galois groups. Motivated by this list, we prove two theorems about the inverse Galois problem for $\mathbb{Q}_p$: one giving a necessary condition for G to be realizable over $\mathbb{Q}_p$ and the other giving a sufficient condition.