C -differential bent functions and perfect nonlinearity

17 USC 105 interim-entered record; under review.The article of record as published may be found at https://doi.org/10.1016/j.dam.2021.10.010Drawing 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).Pantelimon Stănică acknowledges the sabbatical support from Naval Postgraduate School from September 2020 to July 2021

CC-differential bent functions and perfect nonlinearity

Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity and the $c$-differential notion we defined in~\cite{EFRST20}, in this paper we introduce the concept of $c$-differential bent functions in two different ways (thus extending Kumar et al.~\cite{Ku85} classical definition). We further extend the notion of perfect $c$-nonlinear introduced in~\cite{EFRST20}, 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).Comment: 24 page

Further results on constructions of generalized bent Boolean functions

National Natural Science Foundation of China (Grant Nos. 61303263, 61309034)Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS086)China Postdoctoral Science Foundation Funded Project (Grant No. 2015T80600)National Natural Science Foundation of China (Grant Nos. 61303263, 61309034)Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS086)China Postdoctoral Science Foundation Funded Project (Grant No. 2015T80600