Number theory
Algebraic geometry
Information theory
Équipe ATI
The ATI team is part of the Arithmetic, Geometry, Logic and Representations (AGLR) scientific group.
The Arithmetic and Information Theory team was born out of a meeting between researchers working at the interface between number theory, algebraic geometry and their applications to information theory, in particular error-correcting codes and cryptography, thus identifying the three main themes on which the team works. These three themes interact with each other. There is also a theme linked to the regularity theory of completely non-linear elliptic equations, which gave rise to results presented at the International Mathematical Congress in 2010.
Thèmes de recherche
- Algebraic and analytic number theory: Riemann hypothesis, distribution of arithmetic functions, number fields (L functions, zeta functions, number of classes, fundamental units), Frobenius distributions (Lang-Trotter and Sato-Tate conjectures), exponential sums, cyclotomy of Weil sums, diophantine equations.
- Modular shapes. Galois representations.
- Algorithmic number theory: algorithms in finite fields, number fields, function fields and algebraic varieties (Chudnovsky, CM method, Point counting: Schoof-Elkies-Atkin algorithms, p-adic method of Satoh and Mestre by canonical bearing, p-adic method of Kedlaya).
- Study the geometric and arithmetic properties of various geometric objects (curves, abelian varieties, surfaces, etc.), with applications to code theory and cryptography.
- Arithmetic in function fields: turns/suites of function fields defined over finite fields (Kummer, Artin-Schreier, modular, Shimura-type turns), Riemann-Roch spaces, descent problem.
- Number of rational points of algebraic sets, abelian varieties including Prym varieties and Jacobians over finite fields.
- Curves of small kinds.
- Singular curves, optimum and maximum curves.
- Greenhouse obstruction.
- L-adic cohomology.
- Regularity of elliptic and hydrodynamic partial differential equations.
- Computational and effective algebraic geometry.
- Algebraic complexity (complexity of arithmetic operations in finite fields).
- Boolean functions in coding and cryptography, nonlinearity, APN functions and functions with low differential uniformity, asymptotic study of differential uniformity.
- Error-correcting codes, in particular codes on rings and quasi-cyclic codes, interconnection networks, arithmetic networks (lattices).
– Weight of generalized Reed-Muller codes, and codes defined on quadratic or Hermitian varieties. - Construction of corrective codes for information transmission problems using algebraic geometry and number theory. In particular, these are codes with different locality and sparsity conditions.
- Applied cryptography (construction and study of cryptographic primitives and protocols, side channels, pseudo-random generators, secret sharing, authenticated dictionaries).
- Construction of exponentiation algorithms in finite fields.
- Effective implementation of algorithms.
Members:
| ANNI Samuele | MCF AMU |
| AUBRY Yves | MCF Univ. Toulon |
| BALAZARD Michel | CR CNRS |
| BALLET Stéphane | PR AMU |
| BONNECAZE Alexis | PR AMU |
| COLÒ Leonardo | DOCT |
| KOHEL David | PR AMU |
| LOUBOUTIN Stéphane | PR AMU |
| PACIFICO Bastien | DOCT |
| RODIER François | DREM CNRS |
| ROLLAND Robert | MCFEM AMU |
| SOLÉ Patrick | DR CNRS |
| VLADUTS Serge | PR AMU |
| YVON Zoé | DOCT |
On the web: Details on the 2011-2016 activity report – ATI team (page 142-163)
