In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb{F}_2^{n}$ in the Maiorana-McFarland class $\mathcal{MM}$ regarding the origin and cardinality of their $\mathcal{M}$-subspaces, i.e., vector subspaces on which the second-order derivatives of $f$ vanish. By imposing restrictions on permutations $\pi$ of $\mathbb{F}_2^{n/2}$, we specify the conditions, such 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 is invariant under equivalence. Additionally, we give several generic methods of specifying permutations $\pi$ so that $f\in\mathcal{MM}$ 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, one can in a generic manner generate bent functions on $\mathbb{F}_2^{n}$ outside the completed Maiorana-McFarland class $\mathcal{MM}^\#$ 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 two primary classes, the partial spread class $\mathcal{PS}$ and $\mathcal{MM}$. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction, of which size is about $2^{76}$, stems from $\mathcal{PS}$ and $\mathcal{MM}$, whereas the total number of bent functions on $\mathbb{F}_2^8$ is approximately $2^{106}$.

翻译：本文首次系统分析了Maiorana-McFarland类$\mathcal{MM}$中$\mathbb{F}_2^{n}$上弯曲函数$f$的$\mathcal{M}$-子空间（即函数二阶导数消失的向量子空间）的起源与基数。通过对$\mathbb{F}_2^{n/2}$上置换$\pi$施加约束，我们给出了使得Maiorana-McFarland弯曲函数$f(x,y)=x\cdot \pi(y) + h(y)$具有唯一$n/2$维$\mathcal{M}$-子空间的条件。另一方面，我们证明了具有线性结构的置换$\pi$会产生不具有该性质的Maiorana-McFarland弯曲函数。由于$\mathcal{M}$-子空间的数量在等价关系下保持不变，这一研究有助于Maiorana-McFarland弯曲函数的分类。此外，我们给出了若干指定置换$\pi$的通用方法，使得$f\in\mathcal{MM}$具有唯一$\mathcal{M}$-子空间。特别地，利用$\mathcal{M}$-子空间的知识，我们证明了通过对四个适当选取的Maiorana-McFarland弯曲函数进行弯曲4-拼接，可以在任意偶数$n\geq 8$上生成位于完备化Maiorana-McFarland类$\mathcal{MM}^\#$之外的$\mathbb{F}_2^{n}$上弯曲函数。值得注意的是，我们的构造方法能够获得$\mathbb{F}_2^8$上非源自两个主类（部分扩类$\mathcal{PS}$和$\mathcal{MM}$）的不等价弯曲函数。这一成果加深了对八变量弯曲函数起源的理解，因为仅有约$2^{76}$量级的一小部分函数来源于$\mathcal{PS}$和$\mathcal{MM}$类，而$\mathbb{F}_2^8$上弯曲函数的总数约为$2^{106}$。