Explicit infinite families of bent functions outside the completed Maiorana–McFarland class

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 ($\mathcal{M}^{\#}$) 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 $\mathcal{M}^{\#}$ and eventually we obtain many infinite families of bent functions that are provably outside $\mathcal{M}^{\#}$. The fact that a bent function f is in/outside $\mathcal{M}^{\#}$ if and only if its dual is in/outside $\mathcal{M}^{\#}$ 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 (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 $(n−2)$-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 $\mathcal{M}^{\#}$. For instance, for the elementary case of defining a bent function $h(x,y_1,y_2)=f(x)\oplus y_1y_2$ on $\mathbb{F}_2^{n+2}$ using a bent function $f$ on $\mathbb{F}_2^n$, we show that $h$ is outside $\mathcal{M}^{\#}$ if and only if $f$ is outside $\mathcal{M}^{\#}$. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $f_1\parallel f_1\parallel f_2\parallel (1\oplus f_2)$ also gives bent functions outside $\mathcal{M}^{\#}$ if $f_1$ or $f_2$ is outside $\mathcal{M}^{\#}$. 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 outside $\mathcal{M}^{\#}$ are provided.