Vendredi 20 avril 2012 à 14h30 en salle C47

Jean-Pierre Flori (ANSSI)

Titre : Hyper-bent functions with Dillon-like exponents

Résumé :

A few years ago, Charpin and Gong presented a characterization of hyper-bent functions in polynomial form with a certain type of Dillon-like exponents (that is exponents of the form $s (2^m-1)$) : the cyclotomic cosets corresponding to the exponents were supposed to be of maximal size.
Furthermore, another restriction lied on the coefficients of these functions : they had to be defined in $GF(2^m)$ rather than $GF(2^(2m))$.
Using the classical connection between exponential sums and algebraic varieties, Lisonek then reformulated this characterization in terms of hyperelliptic curves.

Subsequently, Mesnager showed how the Charpin--Gong family can be extended to include an additional term with exponent $(2^m+1)/3 (2^m-1)$, without restricition on the corresponding coefficient (which lives in $GF(4)$).
Following this approach, Wang et al. also showed how the Charpin--Gong family can be extended to include an additional term with exponent $(2^m+1)/5 (2^m-1)$, without restricition on the corresponding coefficient (which lives in $GF(16)$).
Similar reformulations in terms of hyperelliptic curves can be deduced.

In this talk, we'll present generalizations of the previous approaches in different directions.
First, we show that the original restriction on the type of Dillon-like in the Charpin--Gong family is superfluous.
Second, we show how to extend the Mesnager approach to an additional term with an arbitrary Dillon-like exponent, thus getting rid of the subfield restriction for the corresponding coefficient.
Finally, we study the practical interest of reformulations in terms of hyperelliptic curves given the current implementations of point counting on hyperelliptic curves.

Remarques : This is joint work with Sihem Mesnager