Almost Maiorana-McFarland bent functions

In this article, we study bent functions on $\mathbb{F}_2^{2m}$ of the form $f(x,y) = x \cdot \phi(y) + h(y)$, where $x \in \mathbb{F}_2^{m-1}$ and $y \in \mathbb{F}_2^{m+1}$, which form the generalized Maiorana-McFarland class (denoted by ${GMM}_{m+1}$) and are referred to as almost Maiorana-McFarland bent functions. We provide a complete characterization of the bent property for such functions and determine their duals. Specifically, we show that $f$ is bent if and only if the mapping $\phi$ partitions $\mathbb{F}_2^{m+1}$ into 2-dimensional affine subspaces, on each of which the function $h$ has odd weight. We investigate which properties of mappings $\phi \colon \mathbb{F}_2^{m+1} \to \mathbb{F}_2^{m-1}$ lead to bent functions of the form $f(x,y) = x \cdot \phi(y) + h(y)$ both inside and outside ${M}^\#$ and provide construction methods for suitable Boolean functions $h$ on $\mathbb{F}_2^{m+1}$. We present a simple algorithm for constructing partitions of the vector space $\mathbb{F}_2^{m+1}$ together with appropriate Boolean functions $h$ that generate bent functions outside ${M}^\#$. When $2m = 8$, we explicitly identify many such partitions that produce at least $2^{78}$ distinct bent functions on $\mathbb{F}_2^8$ that do not belong to ${M}^\#$, thereby generating more bent functions outside ${M}^\#$ than the total number of 8-variable bent functions in ${M}^\#$. Additionally, we demonstrate that concatenating four almost Maiorana-McFarland bent functions outside ${M}^\#$, can result in a bent function ${M}^\#$. This finding answers an open problem posed recently in Kudin et al. (IEEE Trans. Inf. Theory 71(5): 3999-4011, 2025). Conversely, using a similar approach to concatenate four functions each in ${M}^\#$, we generate bent functions that are provably outside ${M}^\#$.