Bent functions satisfying the dual bent condition and permutations with the (Aₘ) property

The concatenation of four Boolean bent functions f=f₁||f₂||f₃||f₄ f = f 1 | | f 2 | | f 3 | | f 4 is bent if and only if the dual bent condition f₁^* + f₂^* + f₃^* + f₄^* =1 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 relations between fᵢ f i are assumed, as well as functions fᵢ f i of a special shape are considered, e.g., fᵢ(x,y)=x· π ᵢ(y)+hᵢ(y) f i ( x , y ) = x · π i ( y ) + h i ( y ) are Maiorana-McFarland bent functions. In the case when permutations π ᵢ π i of F₂ᵐ F 2 m have the (Aₘ) ( A m ) property and Maiorana-McFarland bent functions fᵢ f i satisfy the additional condition f₁+f₂+f₃+f₄=0 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 fᵢ f i explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions fᵢ f i satisfy the condition f₁(x,y)+f₂(x,y)+f₃(x,y)+f₄(x,y)=s(y) 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 (Aₘ) ( 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 f₁,f₂,f₃,f₄ f 1 , f 2 , f 3 , f 4 stemming from the permutations of F₂ᵐ F 2 m with the (Aₘ) ( A m ) property, such that the concatenation f=f₁||f₂||f₃||f₄ f = f 1 | | f 2 | | f 3 | | f 4 does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations π ᵢ π i of F2ᵐ F 2 m with the (Aₘ) ( A m ) property and monomial functions hᵢ h i on F2ᵐ F 2 m , we provide explicit constructions of such bent functions; a particular case of our result shows how one can construct bent functions from APN permutations, when m is odd. 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.