Differential uniformity and second order derivatives for generic polynomials

2017/03/21 by Aubry, Yves, Herbaut, Fabien
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1703.07299

Abstract

For any polynomial f of \mathbb F_2ⁿ[x] we introduce the following characteristic of the distribution of its second order derivative,which extends the differential uniformity notion:δ²(f):=max_\substackα∈ \mathbb F_2^n∗ ,α' ∈ \mathbb F_2^n∗ ,β∈ \mathbb F_2ⁿ α\not=α' \sharp\x∈\mathbb F_2ⁿ | 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.

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