My research focuses on two distinct themes, I worked on successively: group theory and the theory of information transmission.

Group Theory

My work on group theory were mainly the study of

representations of reductive p-adic groups.

• Proof of the theorem of uniqueness of Whittaker models.

I stated a uniqueness property for irreducible admissible representations of a reductive group over a p-adic group and a property of inheritance for representations induced from parabolic subgroups.

• Classification of representations of GSp (4, k), using degenerate Whittaker models

• Spectral decomposition of smooth representations.

I defined a notion of spectral decomposition for these representations, reducing their study to that of sheaves on the dual group. I developed some examples, obtaining results on smooth representations of unipotent groups.

• Calculation of the character of the Steinberg representation.

I showed that the character of Steinberg p-adic reductive group is never zero on the semi-simple elements, in analogy with the case of reductive groups over finite fields.

• Local Integrability of characters of GL (n, k).

• Decomposition of principal unramified representation.

I gave a classification of irreducible components of principal regular representations of split groups and I characterized those which are tempered or square integrable ...

• Decomposition of principal ramified representation (especially for the group GSp (4, k)).

In this paper, I study the irreducible unramified representation of reductive groups over a local field. In the case of GSp (4, k), I determine them all and I study some of their properties.

This article, published in 1988, has recently given rise to minor corrections, following several recent remarks.

Information Theory

My work is currently on error correcting codes and cryptography.

Error correcting codes

• Dimension of Goppa codes

First I improved the lower bound for the dimension of Goppa codes, constructed with a curve and a divisor on this curve. I showed it to be accurate under certain conditions.

• Weight of duals of BCH codes

Then I studied the weights of duals of BCH codes. These codes and their duals are important in coding theory: they are cyclic and are among the first to be developed. Nevertheless, they still pose many problems. I used the fact that they are special Goppa codes.

I first showed that Sloane conjecture on a bound for the minimum distance was incorrect. This study was done using the reduction of certain binary exponential sums under various assumptions.

I have also completed with a student, Eric Férard, further study on an upper bound of exponential sums. With techniques of algebraic geometry, we have improved many of these bounds.

Finally, I found among the duals of BCH codes some codes which are better than all other codes of the same length and dimension, improving minimum distances tables.

• Varieties over finite fields and codes

Following V. Goppa and Y. Manin who showed how to construct codes using algebraic varieties, I tried to apply their methods with varieties having a lot of points to obtain codes with good parameters. The quest for these varieties led me to become interested in Deligne-Lusztig surfaces. I completely determined zeta function, I calculated the canonical divisor and I deduced geometric properties.

I then built the codes from these surfaces using the variety of flags.

I could then apply some of my ideas with a PhD student, Mr F. Edoukou on error correcting codes constructed from hermitian or quadrics varieties.

• Weight of generalized Reed-Muller codes

Finally I studied the determination of the weights of generalized Reed-Muller codes, in collaboration with a PhD student, Mr A. Sboui. We got some of the first weights.

Cryptography

The problem

Boolean functions on the space F2m occur both in the theory of error correcting codes (Reed-Muller codes) and in cryptography (secret key encryption systems).

In both cases, the properties of such systems depend among other things on the nonlinearity of a Boolean function, ie its distance (in Hamming sense) to affine Boolean functions. This is an essential cryptographic key parameter: it controls the resistance to differential and linear attacks of certain cryptosystems such as DES.

It was shown that it was necessary to have Boolean functions with high nonlinearity.

Furthermore, the nonlinearity is related to the covering radius of Reed-Muller codes (one of the oldest open problems in coding theory). This radius is an important parameter codes, particularly for decoding, or in the context of the use of code for the compression of documents.

It is essential, but very difficult to find functions that have a maximum nonlinearity (bent functions that exist only if m is even) or near to the maximum. First in cryptography, in order to increase the number of available functions, in order to better fight against possible attacks, and in coding theory in the case of m odd, in order to approximate the value of the covering radius of Reed and Muller codes.

Results

I paralleled a result of harmonic analysis with an old problem of cryptography, which allowed me to infer many new consequences.

The first result is that almost all Boolean functions in m variables have a non-linearity rather close to the maximum value, a result that I have shown in various settings. Specifically, I showed that almost all Boolean functions in m variables have a nonlinearity of about 2

I studied explicit functions, constructed using the trace of polynomials on F

I also got upper bounds in the number of explicit Boolean functions with nonlinearity given.

Finally, I found a criterion for vector functions are not almost perfect nonlinear (APN). This criterion has permitted to find a number of cases where Boolean functions given by polynomials were not APN for an infinity of finite fields, according to the conjecture that we set out with Y. Aubry and G. McGuire: a polynomial can not be APN for an infinity of fields if it is (CCZ) is equivalent to a monomial of exponent 2