Vectorial Negabent Concepts: Similarities, Differences, and Generalizations

In Pasalic et al., IEEE Trans. Inform. Theory 69 (2023), 2702--2712, and in Anbar, Meidl, Cryptogr. Commun. 10 (2018), 235--249, two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from $\mathbb{F}_2^n$ to the cyclic group $\mathbb{Z}_{2^k}$. It is shown how to obtain nega-$\mathbb{Z}_{2^k}$-bent functions from $\mathbb{Z}_{2^k}$-bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of $\mathbb{Z}_8$-bent functions employing permutations with the $(\mathcal{A}_m)$ property, and more generally we show that the inverse permutation gives rise to $\mathbb{Z}_{2^k}$-bent functions.