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

Abstract

AbstractThe concatenation of four Boolean bent functions $$f=f_1||f_2||f_3||f_4$$ f = f 1 | | f 2 | | f 3 | | f 4 is bent if and only if the dual bent condition $$f_1^* + f_2^* + f_3^* + f_4^* =1$$ 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 $$f_i$$ f i are assumed, as well as functions $$f_i$$ f i of a special shape are considered, e.g., $$f_i(x,y)=x\cdot \pi _i(y)+h_i(y)$$ f i ( x , y ) = x · π i ( y ) + h i ( y ) are Maiorana-McFarland bent functions. In the case when permutations $$\pi _i$$ π i of $$\mathbb {F}_2^m$$ F 2 m have the $$(\mathcal {A}_m)$$ ( A m ) property and Maiorana-McFarland bent functions $$f_i$$ f i satisfy the additional condition $$f_1+f_2+f_3+f_4=0$$ 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 $$f_i$$ f i explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions $$f_i$$ f i satisfy the condition $$f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$$ 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 $$(\mathcal {A}_m)$$ ( 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 $$f_1,f_2,f_3,f_4$$ f 1 , f 2 , f 3 , f 4 stemming from the permutations of $$\mathbb {F}_2^m$$ F 2 m with the $$(\mathcal {A}_m)$$ ( A m ) property, such that the concatenation $$f=f_1||f_2||f_3||f_4$$ f = f 1 | | f 2 | | f 3 | | f 4 does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations $$\pi _i$$ π i of $$\mathbb {F}_{2^m}$$ F 2 m with the $$(\mathcal {A}_m)$$ ( A m ) property and monomial functions $$h_i$$ h i on $$\mathbb {F}_{2^m}$$ 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.