On sets determining the differential spectrum of mappings

The differential uniformity of a mapping F : F2n → F2n is defined as the maximum number of solutions x for equations F (x+a)+F (x) = b when a ̸= 0 and b run over F2n . 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 F2n \ {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 F2n . 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 F2n .