Formula omitted]-differential bent functions and perfect nonlinearity

Abstract

Drawing inspiration from Nyberg’s paper (Nyberg, 1991) on perfect nonlinearity and the c-differential notion we defined in Ellingsen et al. (2020), in this paper we introduce the concept of c-differential bent functions in two different ways (thus extending Kumar et al. (1985) classical definition). We further extend the notion of perfect c-nonlinear introduced in Ellingsen et al. (2020), also in two different ways, and show that, in both cases, the concepts of c-differential bent and perfect c-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functions with these properties are also provided; one such construction provides a large class of PcN functions with respect to all c in some subfield of the field under consideration. We also show that both our classes of 0-differential bents are supersets of permutation polynomials, and that Maiorana–McFarland bent functions are not differential bent (of the first kind).