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 $ {\mathcal {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. While the partition of $\mathbb {F}_{2}^{m+1} $ into 2-dimensional affine subspaces is crucial for the bentness, we demonstrate that the algebraic structure of these subspaces plays an even greater role in ensuring that the constructed bent functions $f$ are excluded from the completed Maiorana-McFarland class $ \mathcal {M}^{\#} $ (the set of bent functions that are extended-affine equivalent to bent functions from the Maiorana-McFarland class $\mathcal {M}$ ). Consequently, 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 $ \mathcal {M}^{\#} $ and provide construction methods for suitable Boolean functions $ h $ on $ \mathbb {F}_{2}^{m+1} $ . As part of this framework, 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 $ \mathcal {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 $ \mathcal {M}^{\#} $ , thereby generating more bent functions outside $ \mathcal {M}^{\#} $ than the total number of 8-variable bent functions in $ \mathcal {M}^{\#} $ (whose cardinality is approximately $2^{77} $ ). Additionally, we demonstrate that concatenating four almost Maiorana-McFarland bent functions outside $ \mathcal {M}^{\#} $ , i.e., defining $ f = f_{1} || f_{2} || f_{3} || f_{4} $ where $ f_{i} \notin \mathcal {M}^{\#} $ , can result in a bent function $ f \in \mathcal {M}^{\#} $ . This finding essentially 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 $ f_{1} || f_{2} || f_{3} || f_{4} $ , where each $ f_{i} \in \mathcal {M}^{\#} $ , we generate bent functions that are provably outside $ \mathcal {M}^{\#} $ .