Differential uniformity and second order derivatives for generic polynomials

Abstract

For any polynomial $f$ of $\mathbb{F}\_2^n[x]$ we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:$${\delta }^2(f):=\max \_\begin{array}{c}\alpha \in \mathbb{F}\_{2^n}^{\ast },{\alpha }^{\prime }\in \mathbb{F}\_{2^n}^{\ast },\beta \in \mathbb{F}\_2^n\alpha \ne {\alpha }^{\prime }\end{array}♯\{x\in \mathbb{F}\_2^n\mid D\_{\alpha ,{\alpha }^{\prime }}^{2}f(x)=\beta \}$$where $D\_{\alpha ,{\alpha }^{\prime }}^{2}f(x):=D\_{\alpha }^{\prime }(D\_\alpha f(x))=f(x)+f(x+\alpha )+f(x+{\alpha }^{\prime })+f(x+\alpha +{\alpha }^{\prime })$ is the second order derivative.Our purpose is to prove a density theorem relative to this quantity,which is an analogue of a density theorem proved by Voloch for the differential uniformity.