%0 Journal Article
%T Differential uniformity and second order derivatives for generic polynomials
%+ Institut de Mathématiques de Toulon - EA 2134 (IMATH)
%+ Institut de Mathématiques de Marseille (I2M)
%+ Université Nice Sophia Antipolis (1965 - 2019) (UNS)
%+ École supérieure du professorat et de l'éducation - Académie de Nice (ESPE Nice)
%A Aubry, Yves
%A Herbaut, Fabien
%< avec comité de lecture
%@ 0022-4049
%J Journal of Pure and Applied Algebra
%I Elsevier
%V 222
%N 5
%P 1095-1110
%8 2018-05
%D 2018
%Z 1703.07299
%R 10.1016/j.jpaa.2017.06.009
%K Differential uniformity
%K Galois closure of a map
%K Chebotarev density theorem.
%Z Mathematics [math]/Algebraic Geometry [math.AG]
%Z Mathematics [math]/Number Theory [math.NT]
%Z Mathematics [math]/Information Theory [math.IT]Journal articles
%X 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:$$\delta^2(f):=\max_{\substack{\alpha \in {\mathbb F}_{2^n}^{\ast} ,\alpha' \in {\mathbb F}_{2^n}^{\ast} ,\beta \in {\mathbb F}_{2^n} \\\alpha\not=\alpha'}} \sharp\{x\in{\mathbb F}_{2^n} \mid D_{\alpha,\alpha'}^2f(x)=\beta\}$$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 theorem relative to this quantity,which is an analogue of a density theorem proved by Voloch for the differential uniformity.
%G English
%2 https://hal.science/hal-01266567/document
%2 https://hal.science/hal-01266567/file/Aubry_Herbaut_170320.pdf
%L hal-01266567
%U https://hal.science/hal-01266567
%~ UNICE
%~ UNIV-TLN
%~ CNRS
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%~ EC-MARSEILLE
%~ INSMI
%~ I2M
%~ I2M-2014-
%~ IMATH
%~ UNIV-COTEDAZUR
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