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Contemporary Mathematics
Differentially 4uniform functions
Yves Aubry and Fran¸cois Rodier
Abstract. We give a geometric characterization of vectorial Boolean func
tions with differential uniformity 4. This enables us to give a necessary
condition on the degree of the base field for a function of degree 2r  1 to be
differentially 4uniform.
1. Introduction
We are interested in vectorial Boolean functions from the F2vectorial space Fm
2
to itself in m variables, viewed as polynomial functions f : F2m  F2m over the
field F2m in one variable of degree at most 2m
1. For a function f : F2m  F2m ,
we consider, after K. Nyberg (see [16]), its differential uniformity
(f) = max
=0,
{x F2m  f(x + ) + f(x) = }.
This is clearly a strictly positive even integer.
Functions f with small (f) have applications in cryptography (see [16]). Such
