When does a bent concatenation not belong to the completed Maiorana-McFarland class?

TL;DR

This paper characterizes when certain Boolean bent function concatenations do not belong to the completed Maiorana-McFarland class, providing conditions and construction methods for such functions.

Contribution

It offers a complete characterization of the structure of -spaces for specific bent function concatenations and identifies conditions for functions to lie outside the Maiorana-McFarland class.

Findings

Full characterization of -spaces for concatenations of two and four functions.

Necessary and sufficient conditions for a concatenated bent function to be outside .

Explicit construction methods for bent functions outside the Maiorana-McFarland class.

Abstract

Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation $f$ (not) belong to the completed Maiorana-McFarland class $\mathcal{M}^\#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-subspaces for the concatenation of the form $f=f_1||f_2$ and $f=f_1||f_2||f_3||f_4$, which allows us to specify the necessary and sufficient conditions so that $f$ is outside $\mathcal{M}^\#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^\#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.