Abstract
Bent functions \(f: V_{n}\rightarrow \mathbb {F}_{p}\) play an important role in constructing partial difference sets, where \(V_{n}\) denotes an n-dimensional vector space over \(\mathbb {F}_{p}\), p is an odd prime. In [2, 3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et al. showed that for certain vectorial dual-bent functions \(F: V_{n}\rightarrow V_{s}\), the preimage set of 0 for F forms a partial difference set. In [3], Çeşmelioğlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) for F forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for certain vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) for F and the preimage set of any coset of some subgroup of \(\mathbb {F}_{p^s}^{*}\) for F form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non-)quadratic vectorial dual-bent functions. In this paper, we illustrate that many results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results. In [2], the authors considered weakly regular p-ary bent functions f with \(f(0)=0\). They showed that if such a function f is an l-form with \(gcd(l-1, p-1)=1\) for some integer \(1\le l\le p-1\), then f is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in [3].
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Acknowledgements
This research is supported by the National Key Research and Development Program of China (Grant No. 2018YFA0704703), the National Natural Science Foundation of China (Grant Nos. 12141108 and 61971243), the Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China (Nankai University), and the Nankai Zhide Foundation. The authors would like to thank the two anonymous reviewers and the Associate Editor for their valuable suggestions and comments that helped to greatly improve the paper.
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Wang, J., Fu, FW. New results on vectorial dual-bent functions and partial difference sets. Des. Codes Cryptogr. 91, 127–149 (2023). https://doi.org/10.1007/s10623-022-01103-6
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DOI: https://doi.org/10.1007/s10623-022-01103-6
Keywords
- Bent functions
- Vectorial bent functions
- Vectorial dual-bent functions
- Partial difference set
- Walsh transform