Conference Agenda

Polynomial equations are central in algebraic geometry, being algebraic varieties geometric manifestations of solutions of systems of polynomial equations. Actually, modern algebraic geometry is based on the use of techniques for studying and solving geometrical problems about these sets of zeros. At the same time, polynomial equations have found interesting applications in coding theory and cryptography. The interplay between algebraic geometry and coding theory is old and goes back to the first examples of algebraic codes defined with polynomials and codes coming from algebraic curves. More recently, polynomial equations have found important applications in cryptography as well. For example, in multivariate cryptography, one of the prominent candidates for post-quantum cryptosystems, the trapdoor one-way function takes the form of a multivariate quadratic polynomial map over a finite field. Furthermore, the efficiency of the index calculus attack to break an elliptic curve cryptosystem relies on the effectiveness of solving a system of multivariate polynomial equations. This session will feature recent progress in these and other applications of polynomial equations to coding theory and cryptography.

To each linear code defined over a finite field one can define its associated matroid and its generalized Hamming weights which are the same as those of the code. Johnsen and Verdure. showed that the generalized Hamming weights of a matroid are determined by the graded Betti numbers of the Stanley-Reisner ring of the simplicial complex whose faces are the independent set of M. In this talk we go a step further: our practical results indicate that the generalized Hamming weights of a linearcode can be obtained from the monomial ideal associated with a test-set for the code.

In this talk, we consider the problem of constructing codes with good parameters from algebraic surfaces. We start from two constatations. The first one, due to Voloch and Zarzar in a 2007 article, is that surfaces with a small Picard rank, in particular those with Picard rank one, seem to be interesting candidates to provide good codes. The second one, is that several nice examples in the literature of surfaces yielding good codes can be understood in a unified context : that of del Pezzo surfaces.

We will study the classification of del Pezzo surfaces over finite fields and consider their anticanonical codes. Such surfaces can be classified by the action of the Frobenius on the (geometric) Picard lattice, which gives many properties such as the (arithmetic) Picard number or the number of rational points. This rich structure of del Pezzo surfaces permits to obtain fine estimates of the parameters of these codes and even to compute their automorphism groups in some cases. This investigation led to the discovery of new codes whose parameters beat the best known codes listed in the database codetables.de.

This is a collaboration with Blache, Hallouin, Madore, Nardi, Rambaud and Randriam.

We give a new perspective on extremal codes. We obtain upper and lower bounds on the density functions of a number of families of codes within a larger family. In many cases, these bounds have expressions involving polynomials in indeterminate q, where q is the size of the underlying scalar field.

We use these expressions to obtain precise asymptotic estimates of these quantities and hence the density functions for some families of codes possessed of a particular extremal property. We introduce the idea of a partition-balanced family of codes, and show how the combinatorial invariants of such families can be used to obtain estimates on the number of codes satisfying a particular property.

In particular, we show that the MRD matrix codes are not dense in the family of all matrix codes of a given fixed dimension, unlike the MRD vector codes.

In this talk, we will present the designs of multivariate signatures. The focus will be on the schemes submitted to the 2017 NIST post-quantum standard submissions. We will present the key security analysis tools and the main challenges for these schemes.