On sets determining the differential spectrum of mappings

Special issue on the honor of Gerard CohenInternational audienceThe differential uniformity of a mapping $F : F 2 n → F 2 n$ is defined as the maximum number of solutions $x$ for equations $F (x+a)+F (x) = b$ when $a ̸ = 0$ and $b$ run over $F 2 n$. In this paper we study the question whether it is possible to determine the differential uniformity of a mapping by considering not all elements $a ̸ = 0$, but only those from a special proper subset of $F 2 n \ {0}$. We show that the answer is " yes " , when $F$ has differential uniformity 2, that is if $F$ is APN. In this case it is enough to take $a ̸ = 0$ on a hyperplane in $F 2 n$. Further we show that also for a large family of mappings F of a special shape, it is enough to consider a from a suitable multiplicative subgroup of $F 2 n$