Differentially 4-uniform functions

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

Nonlinarity of Boolean functions and hyperelliptic curves

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent results of Maisner and Nart about zeta functions of supersingular curves of genus 2. We give some criterion for a vectorial function not to be almost perfect nonlinear

Bounds on the degree of APN polynomials The Case of x−1+g(x)x^{-1}+g(x)

We prove that functions f:\f{2^m} \to \f{2^m} of the form $f(x)=x^{-1}+g(x)$ where $g$ is any non-affine polynomial are APN on at most a finite number of fields \f{2^m}. Furthermore we prove that when the degree of $g$ is less then 7 such functions are APN only if $m \le 3$ where these functions are equivalent to $x^3$

Sur la non-linéarité des fonctions booléennes

Boolean functions on the space $F_{2}^m$ are not only important in the theory of error-correcting codes, but also in cryptography, wherethey occur in private key systems. In these two cases, the nonlinearity ofthese function is a main concept. In this article, I show that the spectral amplitude of booleanfunctions, which is linked to their nonlinearity, is of theorder of $2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by$2^{m/2}$ and$2^m$.Moreover I examine a conjecture of Patterson and Wiedemann saying that theminimum of this spectral amplitude is as close as desired to $2^{m/2}$.I also study a weaker conjecture about the moments of order 4 of theirFourier transform. This article is inspired by works of Salem, Zygmund,Kahane and others about the related problem of real polynomials withrandom coefficients

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

Functions of degree 4e that are not APN infinitely often

International audienceWe prove a necessary condition for some polynomials of degree 4e (e an odd number) to be APN over F q n for large n, and we investigate the polynomials f of degree 12