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The concatenation of four Boolean bent functions $f=f_1||f_2||f_3||f_4$ isbent 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 dualitycondition is in general quite a difficult task. Commonly, to simplify thisproblem, 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-McFarlandbent functions $f_i$ satisfy the additional condition $f_1+f_2+f_3+f_4=0$, thedual bent condition is known to have a relatively simple shape allowing tospecify the functions $f_i$ explicitly. In this paper, we generalize thisresult for the case when Maiorana-McFarland bent functions $f_i$ satisfy thecondition $f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$ and provide a constructionof new permutations with the $(\mathcal{A}_m)$ property from the old ones.Combining these two results, we obtain a recursive construction method of bentfunctions satisfying the dual bent condition. Moreover, we provide a genericcondition on the Maiorana-McFarland bent functions stemming from thepermutations of $\mathbb{F}_2^m$ with the $(\mathcal{A}_m)$ property, such thattheir concatenation does not belong, up to equivalence, to theMaiorana-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 bentfunctions. Finally, with our construction method, we explain how one canconstruct homogeneous cubic bent functions, noticing that only very few designmethods of these objects are known.

Bent functions satisfying the dual bent condition and permutations with the $(\mathcal{A}_m)$ property