Conference "Algorithms and Arithmetic"
with applications to cryptography

General Information

The French-Russian Poncelet Laboratory organizes a conference on "Algorithms and Arithmetic", Mai 17-19, 2010, to take place in the Independent University of Moscow.

Regarding the recent development of information technology, algorithms are nowadays important tools which bridge questions in various mathematical areas, and, especially number theory, with applications in everyday life. One example is of course cryptography which is becoming today a more and more important topic and whose main goal is to develop algorithms to solve natural questions in arithmetic (such as factorization or discrete logarithm problem). Nevertheless it would be unfair to limit algorithms in number theory just to cryptography. Just regarding any symbolic computation software (like Pari, Magma, Sage), one can see that there exists a real desire to manipulate on computers a large family of mathematical objects (sometimes very abstract) coming from number theory: number fields, p-adic numbers, Galois groups, schemes, modular forms, etc. Of course, such tools are very important also for theorists for instance because they help them everyday to easily test conjectures.

The mail goal of this conference is to give an overview of the role of algorithmic in current research in arithmetic. Since it is of course not reasonable to address all subjects in just one week, we have decided to only focus on the two following ones.

• Factorization of integers and discrete logarithm problem (on the multiplicative group of finite fields and also on groups of rational points of jacobians).
• Galois Theory. In particular, we will talk about effective methods to compute Galois groups of rationnal irreducible polynomials (that is Galois groups of number fields), inverse Galois Problem, and p-adic Galois representations (in relation with modular forms and cohomology).

For each subject, we plan to devote one half-day (or, if necessary, one day) to survey talks. Several research talks will then follow.

The conference benefited from the support of the EADS Foundation Chair, the Russian Foundation for Basic Research, Centre National de la Recherche Scientifique. It is organized within the framework of the year of French-Russian cooperation.