Vectorial Negabent Concepts: Similarities, Differences, and Generalizations
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arXiv
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2024
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| author | Anbar, Nurdagül Kudin, Sadmir Meidl, Wilfried Pasalic, Enes Polujan, Alexandr |
| author_facet | Anbar, Nurdagül Kudin, Sadmir Meidl, Wilfried Pasalic, Enes Polujan, Alexandr |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_05677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vectorial Negabent Concepts: Similarities, Differences, and Generalizations Anbar, Nurdagül Kudin, Sadmir Meidl, Wilfried Pasalic, Enes Polujan, Alexandr Combinatorics Discrete Mathematics Information Theory 05B10 and 06E30 and 14G50 and 94C30 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. |
| title | Vectorial Negabent Concepts: Similarities, Differences, and Generalizations |
| topic | Combinatorics Discrete Mathematics Information Theory 05B10 and 06E30 and 14G50 and 94C30 |
| url | https://arxiv.org/abs/2402.05677 |