Paper 2022/1126

Explicit infinite families of bent functions outside $\mathcal{MM}^\#$

Enes Pasalic
Amar Bapić, University of Primorska
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 ($\mathcal{MM}^\#$) 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.

Available format(s)
Publication info
4-decomposition Class inclusion 5-valued functions Bent functions Dual functions Plateaued functions Walsh support
Contact author(s)
amar bapic @ famnit upr si
2022-08-31: approved
2022-08-30: received
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      author = {Enes Pasalic and Amar Bapić and Fengrong Zhang and Yongzhuang Wei},
      title = {Explicit infinite families of   bent functions outside $\mathcal{MM}^\#$},
      howpublished = {Cryptology ePrint Archive, Paper 2022/1126},
      year = {2022},
      note = {\url{}},
      url = {}
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