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For any polynomial $f$ of ${\mathbb F}\_{2^n}[x]$ we introduce the followingcharacteristic of the distribution of its second order derivative,which extendsthe differential uniformity notion:$$\delta^2(f):=\max\_{\substack{\alpha \in{\mathbb F}\_{2^n}^{\ast} ,\alpha' \in {\mathbb F}\_{2^n}^{\ast} ,\beta \in{\mathbb F}\_{2^n} \alpha\not=\alpha'}} \sharp\{x\in{\mathbb F}\_{2^n} \midD\_{\alpha,\alpha'}^2f(x)=\beta\}$$where$D\_{\alpha,\alpha'}^2f(x):=D\_{\alpha'}(D\_{\alpha}f(x))=f(x)+f(x+\alpha)+f(x+\alpha')+f(x+\alpha+\alpha')$is the second order derivative.Our purpose is to prove a density theoremrelative to this quantity,which is an analogue of a density theorem proved byVoloch for the differential uniformity.

Differential uniformity and second order derivatives for generic polynomials