Paper 2022/1126

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

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.