On sets determining the differential spectrum of mappings

Abstract

The 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 .

Differential uniformity and the associated codes of cryptographic functions

“…It is well-known that F is APN if and only if all its derivatives are 2-to-1. Actually, it is sufficient that this last property holds for 2 n−1 − 1 well-chosen derivatives [10]: F is APN if and only if D a F is 2-to-1 for all non-zero a of any hyperplane of F 2 n . We will show that this result can be generalized, by considering the derivatives in respect with subspaces of smaller dimensions.…”

“…By Lemma 3.2, we know that the codewords of weight 3 and 4 are just all the elements of the setS(F ) = (a,b)∈F * 2 n ×F 2 n S(T a,b )(see(10) and(12) for the definitions). Every set S(T a,b ) is of size κ a,b 2 , where κ a,b is the size of T a,b .…”