Section: Partnerships and Cooperations

National Initiatives

ANR AlgoL : Algorithmics of $L$-functions

Participants : Bill Allombert, Karim Belabas, Henri Cohen, Jean-Marc Couveignes, Andreas Enge.

http://www.math.u-bordeaux1.fr/~belabas/algol/index.html

The AlgoL project comprises research teams in Bordeaux, Montpellier, Lyon, Toulouse and Besançon.

It studies the so-called $L$-functions in number theory from an algorithmic and experimental point of view. $L$-functions encode delicate arithmetic information, and crucial arithmetic conjectures revolve around them: Riemann Hypotheses, Birch and Swinnerton-Dyer conjecture, Stark conjectures, Bloch-Kato conjectures, etc.

Most of current number theory conjectures originate from (usually mechanised) computations, and have been thoroughly checked numerically. $L$-functions and their special values are no exception, but available tools and actual computations become increasingly scarce as one goes further away from Dirichlet $L$-functions. We develop theoretical algorithms and practical tools to study and experiment with (suitable classes of) complex or $p$-adic $L$-functions, their coefficients, special or general values, and zeroes. For instance, it is not known whether $K$-theoretic invariants conjecturally attached to special values are computable in any reasonable complexity model. On the other hand, special values are often readily computed and sometimes provide, albeit conjecturally, the only concrete handle on said invariants.

New theoretical results are translated into new or more efficient functions in the Pari/Gp system.

The project lasted from 15/11/2007 to 15/02/2012, for 51 months it received an ANR funding of 200k€ for a global cost of 1M€.

ANR Peace – Parameter spaces for Efficient Arithmetic and Curve security Evaluation

Participants : Bill Allombert, Karim Belabas, Jean-Marc Couveignes, Andreas Enge, Nicolas Mascot, Enea Milio, Aurel Page, Damien Robert.

http://chic2.gforge.inria.fr/

The Peace project is joint between the research teams of Institut de Recherche en Mathématiques de Rennes (IRMAR), Lfant and Institut Mathématiques de Luminy (IML).

The project aims to constitute a comprehensive and coherent approach towards a better understanding of theoretical and algorithmic aspects of the discrete logarithm problem on algebraic curves of small genus. On the theorical side, this includes an effective description of moduli spaces of curves, of abelian varieties, the maps that link these spaces and the objects they classify. The effective manipulation of moduli objects will allow us to develop a better understanding of the algorithmic difficulty of the discrete logarithm problem on curves, which may have dramatic consequences on the security and efficiency of already deployed cryptographic devices.

One of the anticipated outcomes of this proposal is a new set of general criteria for selecting and validating cryptographically secure curves (or families of curves) suitable for use in cryptography. Instead of publishing fixed curves, as is done in most standards, we aim at proposing generating rationales along with explicit theoretical and algorithmic criteria for their validation.

ANR Simpatic – SIM and PAiring Theory for Information and Communications security

Participant : Damien Robert.

The Simpatic project is an industrial research project, formed by academic research teams and industrial partners: Orange Labs, École Normale Supérieure, INVIA, Oberthur Technologies, ST-Ericsson France, Université de Bordeaux 1, Université de Caen Basse-Normandie, University of Paris 8.

The aim of the Simpatic project is to provide the most possible efficient and secure hardware/software implementation of a bilinear pairing in a SIM card. This implementation will then be used to improve and develop new cryptographic efficient algorithms and protocols in the context of mobile phones and SIM cards. The project will more precisely focus on e-ticketing and e-cash, on cloud storage and on the security of contactless and of remote payment systems.

As a member, Damien Robert will aim to bridge the gap between the theoretical results described in the pairing module and the practical realisation of pairing-based SIM cards in an industrial setting.