Differential uniformity and second order derivatives for generic polynomials
Year: 2017
Type: preprint
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:$$δ^2(f):=\max\_{\substack{α\in {\mathbb F}\_{2^n}^{\ast} ,α' \in {\mathbb F}\_{2^n}^{\ast} ,β\in {\mathbb F}\_{2^n} α\not=α'}} \sharp\{x\in{\mathbb F}\_{2^n} \mid D\_{α,α'}^2f(x)=β\}$$where $D\_{α,α'}^2f(x):=D\_{α'}(D\_αf(x))=f(x)+f(x+α)+f(x+α')+f(x+α+α')$ 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.
Source: arXiv (Cornell University)
Authors: Aubry, Yves, Herbaut, Fabien
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2951180803