Vectorial Negabent Concepts: Similarities, Differences, and Generalizations
Nurdagül Anbar, Sadmir Kudin, Wilfried Meidl, Enes Pasalic, Alexandr Polujan
TL;DR
The work reconciles two vectorial negabent notions and generalizes negabent to generalized Boolean functions valued in $\mathbb{Z}_{2^k}$, linking nega-$\mathbb{Z}_{2^k}$-bent and gbent concepts through group-theoretic and spectral characterizations. It introduces the nega-$\mathbb{Z}_{2^k}$-Hadamard transform derived from a tailored group structure and provides equivalent conditions for nega-bent and nega-gbent functions, connecting them to derivative-balancing and relative difference sets. A primary contribution is a construction framework using permutations with the $(\mathcal{A}_m)$ property to generate gbent and $\mathbb{Z}_{2^k}$-bent functions via Maiorana-McFarland forms, including explicit $\mathbb{Z}_8$-bent instances and infinite families. The paper also discusses the limitations of methods based on complete permutations for higher $k$ and odd dimensions, and highlights open questions about simultaneous vectorial bent-negabent and gbent realizations. These results expand the design toolbox for generalized bent structures with potential applications in CDMA, quantum contexts, and combinatorial design theory.
Abstract
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.