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arXiv · 2006.12535

$C$-differential bent functions and perfect nonlinearity

Pantelimon Stanica·Sugata Gangopadhyay·Aaron Geary·Constanza Riera·Anton Tkachenko
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Abstract

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).

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Pantelimon Stanica, Sugata Gangopadhyay, Aaron Geary, Constanza Riera, Anton Tkachenko. 2020-06-22. $C$-differential bent functions and perfect nonlinearity. https://arxiv.org/abs/2006.12535

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