On the existence and non-existence of some classes of bent–negabent functions

In this paper we investigate different questions related to bent–negabent functions. We first take an expository look at the state-of-the-art research in this domain and point out some technical flaws in certain results and fix some of them. Further, we derive a necessary and sufficient condition for which the functions of the form \({\mathbf{x}}\cdot \pi ({\mathbf{y}})\oplus h({\mathbf{y}})\) [Maiorana–McFarland (\({\mathcal {M}}\))] is bent–negabent, and more generally, we study the non-existence of bent–negabent functions in the \({\mathcal {M}}\) class. We also identify some functions that are bent–negabent. Next, we continue the recent work by Mandal et al. (Discrete Appl Math 236:1–6, 2018) on rotation symmetric bent–negabent functions and show their non-existence in larger classes. For example, we prove that there is no rotation symmetric bent–negabent function in \(4p^k\) variables, where p is an odd prime. We present the non-existence of such functions in certain classes that are affine transformations of rotation symmetric functions. Keeping in mind the existing literature, we correct here some technical issues and errors found in other papers and provide some novel results.