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Title:
Explicit infinite families of bent functions outside \mathcal{MM}^\#
Authors: Enes Pasalic, Amar Bapić, Fengrong Zhang, Yongzhuang Wei
Abstract: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 (\mathcal{MM}^\#) 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 in \mathcal{MM}^\# and eventually we obtain many infinite families of bent functions that are provably outside \mathcal{MM}^\#. The fact that a bent function f is in/outside \mathcal{MM}^\# if and only if its dual is in/outside \mathcal{MM}^\# is employed in the so-called 4-decomposition of a bent function on \mathbb{F}_2^n, which was originally considered by Canteaut and Charpin \cite{Decom} in terms of the second-order derivatives and later reformulated in \cite{HPZ2019} in terms of the duals of its restrictions to the cosets of an (n-2)-dimensional subspace V. 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 \mathcal{MM}^\#. For instance, for the elementary case of defining a bent function h(\mathbf{x},y_1,y_2)=f(\mathbf{x}) \oplus y_1y_2 on \mathbb{F}_2^{n+2} using a bent function f on \mathbb{F}_2^n, we show that h is outside \mathcal{MM}^\# if and only if f is outside \mathcal{MM}^\#. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation f_1||f_1||f_2||(1\oplus f_2) also gives bent functions outside \mathcal{MM}^\# if either f_1 or f_2 is outside \mathcal{MM}^\#. 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 designing infinite families of bent functions outside \mathcal{MM}^\#, using the spectral domain design are proposed.
ePrint: https://eprint.iacr.org/2022/1126
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