Differentially 4-uniform functions

We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4

A few more functions that are not APN infinitely often

We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold member ($2^k+1$) or a Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of even degree, and functions that have degree $2^k+1$

A new large class of functions not APN infinitely often

In this paper, we show that there is no vectorial Boolean function of degree 4e, with e satisfaying certain conditions, which is APN over infinitely many extensions of its field of definition. It is a new step in the proof of the conjecture of Aubry, McGuire and Rodie

Borne sur le degré des polynômes presque parfaitement non-linéaires

19 pagesThe vectorial Boolean functions are employed in cryptography to build block coding algorithms. An important criterion on these functions is their resistance to the differential cryptanalysis. Nyberg defined the notion of almost perfect non-linearity (APN) to study resistance to the differential attacks. Up to now, the study of functions APN was especially devoted to power functions. Recently, Budaghyan and al. showed that certain quadratic polynomials were APN. Here, we will give a criterion so that a function is not almost perfectly non-linear. H. Janwa showed, by using Weil's bound, that certain cyclic codes could not correct two errors. A. Canteaut showed by using the same method that the functions powers were not APN for a too large value of the exponent. We use Lang and Weil's bound and a result of P. Deligne on the Weil's conjectures (or more exactly improvements given by Ghorpade and Lachaud) about surfaces on finite fields to generalize this result to all the polynomials. We show therefore that a polynomial cannot be APN if its degree is too large

A direct proof of APN-ness of the Kasami functions

Using recent results on solving the equation $X^{2^k+1}+X+a=0$ over a finite field $\mathbb{F}_{2^n}$, we address an open question raised by the first author in WAIFI 2014 concerning the APN-ness of the Kasami functions $x\mapsto x^{2^{2k}-2^k+1}$ with $gcd(k,n)=1$, $x\in\mathbb{F}_{2^n}$