Vectorial Negabent Concepts: Similarities, Differences, and Generalizations

TL;DR

This paper clarifies the differences and similarities between two vectorial negabent concepts, extends negabent functions to generalized Boolean functions over cyclic groups, and presents new constructions for bent functions using permutations.

Contribution

It distinguishes between two negabent concepts, extends the negabent notion to functions over \\mathbb{Z}_{2^k}, and introduces permutation-based constructions for bent functions.

Findings

Clarified the relationship between two negabent concepts.

Extended negabent functions to generalized Boolean functions over cyclic groups.

Provided new constructions of \\mathbb{Z}_8-bent functions using permutations.

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.