Paper 2022/1126

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

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 ($$) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in $$ and eventually we obtain many infinite families of bent functions that are provably outside $$. The fact that a bent function $$ is in/outside $$ if and only if its dual is in/outside $$ is employed in the so-called 4-decomposition of a bent function on $$, which was originally considered by Canteaut and Charpin \cite{Decom} in terms of the second-order derivatives and later reformulated in \cite{HPZ2019} in terms of the duals of its restrictions to the cosets of an $$-dimensional subspace $$. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside $$. For instance, for the elementary case of defining a bent function $$ on $$ using a bent function $$ on $$, we show that $$ is outside $$ if and only if $$ is outside $$. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $$ also gives bent functions outside $$ if either $$ or $$ is outside $$. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of designing infinite families of bent functions outside $$, using the spectral domain design are proposed.