The concatenation of four Boolean bent functions $f=f_1||f_2||f_3||f_4$ is bent if and only if the dual bent condition $f_1^* + f_2^* + f_3^* + f_4^* =1$ is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain connections between $f_i$ are assumed, as well as functions $f_i$ of a special shape are considered, e.g., $f_i(x,y)=x\cdot\pi_i(y)+h_i(y)$ are Maiorana-McFarland bent functions. In the case when permutations $\pi_i$ of $\mathbb{F}_2^m$ have the $(\mathcal{A}_m)$ property and Maiorana-McFarland bent functions $f_i$ satisfy the additional condition $f_1+f_2+f_3+f_4=0$, the dual bent condition is known to have a relatively simple shape allowing to specify the functions $f_i$ explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions $f_i$ satisfy the condition $f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$ and provide a construction of new permutations with the $(\mathcal{A}_m)$ property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions stemming from the permutations of $\mathbb{F}_2^m$ with the $(\mathcal{A}_m)$ property, such that their concatenation does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations $\pi_i$ of $\mathbb{F}_{2^m}$ with the $(\mathcal{A}_m)$ property and monomial functions $h_i$ on $\mathbb{F}_{2^m}$, we provide explicit constructions of such bent functions. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.

翻译：四个布尔弯曲函数$f=f_1||f_2||f_3||f_4$的级联是弯曲函数当且仅当对偶弯曲条件$f_1^* + f_2^* + f_3^* + f_4^* =1$成立。然而，明确给出满足该对偶条件的四个弯曲函数通常是一项相当困难的任务。为解决这一问题，通常假设$f_i$之间存在特定关联，并考虑具有特殊形式的函数$f_i$，例如$f_i(x,y)=x\cdot\pi_i(y)+h_i(y)$为Maiorana-McFarland弯曲函数。当$\mathbb{F}_2^m$上的置换$\pi_i$具有$(\mathcal{A}_m)$性质且Maiorana-McFarland弯曲函数$f_i$满足附加条件$f_1+f_2+f_3+f_4=0$时，已知对偶弯曲条件具有相对简单的形式，从而能显式确定函数$f_i$。本文将该结果推广至Maiorana-McFarland弯曲函数$f_i$满足条件$f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$的情形，并基于已有置换构造了具有$(\mathcal{A}_m)$性质的新置换。结合这两项结果，我们得到了满足对偶弯曲条件的弯曲函数的递归构造方法。此外，我们给出了来源于具有$(\mathcal{A}_m)$性质的$\mathbb{F}_2^m$置换的Maiorana-McFarland弯曲函数的一般条件，使得这些函数的级联在等价意义下不属于Maiorana-McFarland类。利用$\mathbb{F}_{2^m}$上具有$(\mathcal{A}_m)$性质的单项置换$\pi_i$和$\mathbb{F}_{2^m}$上的单项函数$h_i$，我们给出了此类弯曲函数的显式构造。最后，通过我们的构造方法，阐述了如何构造齐次三次弯曲函数——注意到这类对象目前仅有极少的已知设计方法。