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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: $$\delta^2(f):=\max_{\substack{ \alpha \in {\mathbb F}_{2^n}^{\ast} ,\alpha' \in {\mathbb F}_{2^n}^{\ast} ,\beta \in {\mathbb F}_{2^n} \\ \alpha\not=\alpha'}} \sharp\{x\in{\mathbb F}_{2^n} \mid D_{\alpha,\alpha'}^2f(x)=\beta\}$$ 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.
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https://hal.archives-ouvertes.fr/hal-01266567
Contributor : Yves Aubry Connect in order to contact the contributor
Submitted on : Monday, March 20, 2017 - 1:15:11 PM
Last modification on : Wednesday, September 14, 2022 - 5:14:26 PM
Long-term archiving on: : Wednesday, June 21, 2017 - 12:52:25 PM

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Yves Aubry, Fabien Herbaut. Differential uniformity and second order derivatives for generic polynomials. Journal of Pure and Applied Algebra, 2018, 222 (5), pp.1095-1110. ⟨10.1016/j.jpaa.2017.06.009⟩. ⟨hal-01266567⟩

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