Explicit infinite families of bent functions outside the completed Maiorana–McFarland class
Explicit infinite families of bent functions outside the completed Maiorana–McFarland class
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DOI：
10.1007/s10623-023-01204-w
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
E. Pasalic;A. Bapić;Fengrong Zhang;Yongzhuang Wei
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文献类型：
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作者：
E. Pasalic;A. Bapić;Fengrong Zhang;Yongzhuang Wei
During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland () class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion inand eventually we obtain many infinite families of bent functions that are provably outside. The fact that a bent functionfis in/outsideif and only if its dual is in/outsideis employed in the so-called 4-decomposition of a bent function on, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an-dimensional subspaceV. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside. For instance, for the elementary case of defining a bent functiononusing a bent functionfon, we show thathis outsideif and only iffis outside. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenationalso gives bent functions outsideiforis outside. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outsideare provided.
During the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland () class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion inand eventually we obtain many infinite families of bent functions that are provably outside. The fact that a bent functionfis in/outsideif and only if its dual is in/outsideis employed in the so-called 4-decomposition of a bent function on, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an-dimensional subspaceV. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside. For instance, for the elementary case of defining a bent functiononusing a bent functionfon, we show thathis outsideif and only iffis outside. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenationalso gives bent functions outsideiforis outside. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outsideare provided.