Non-linearity of the Carlet-Feng function, and repartition of Gauss sums

The search for Boolean functions that can withstand the main crypyographic attacks is essential. In 2008, Carlet and Feng studied a class of functions which have optimal cryptographic properties with the exception of nonlinearity for which they give a good but not optimal bound. Carlet and some people who have also worked on this problem of nonlinearity have asked for a new answer to this problem. We provide a new solution to improve the evaluation of the nonlinearity of the Carlet-Feng function, by means of the estimation of the distribution of Gauss sums. This work is in progress and we give some suggestions to improve this work

Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < \epsilon<2$, any function with algebraic thickness $n^{3-\epsilon}$ is constant on some affine subspace of dimension $\Omega\left(n^{\frac{\epsilon}{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $\Theta(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $\Theta(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $\Omega\left(2^{n^{1/6}}\right)$.Comment: Final version published in FCT'201

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