Design and Analysis of Bent Functions Using M -Subspaces

In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb {F}_{2}^{n}$ in the Maiorana-McFarland class $\mathcal {M}$ regarding the origin and cardinality of their $\mathcal {M}$ -subspaces, i.e., vector subspaces such that for any two elements $a,b$ from this subspace, the second-order derivative $D_{a}D_{b}f$ is the zero function on $\mathbb {F}_{2}^{n}$ . By imposing restrictions on permutations $\pi $ of $\mathbb {F}_{2}^{n/2}$ , we specify the conditions so that Maiorana-McFarland bent functions $f(x,y)=x\cdot \pi (y) + h(y)$ admit a unique $\mathcal {M}$ -subspace of dimension $n/2$ . On the other hand, we show that permutations $\pi $ with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of $\mathcal {M}$ -subspaces of a fixed dimension is invariant under equivalence. Additionally, we give several generic methods of specifying permutations $\pi $ so that $f\in \mathcal {M}$ admits a unique $\mathcal {M}$ -subspace. Most notably, using the knowledge about $\mathcal {M}$ -subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions on $\mathbb {F}_{2}^{n-2}$ , one can in a generic manner generate bent functions on $\mathbb {F}_{2}^{n}$ outside the completed Maiorana-McFarland class $\mathcal {M}^{\#}$ for any even $n\geq 8$ . Remarkably, with our construction methods, it is possible to obtain inequivalent bent functions on $\mathbb {F}_{2}^{8}$ not stemming from the two primary classes, the partial spread class $\mathcal {PS}$ and $\mathcal {M}$ . In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction of about 276 bent functions stems from $\mathcal {PS}$ and $\mathcal {M}$ , whereas their total number on $\mathbb {F}_{2}^{8}$ is approximately 2106.