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

Abstract

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 relations 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 \(f_1,f_2,f_3,f_4\) stemming from the permutations of \(\mathbb {F}_2^m\) with the \((\mathcal {A}_m)\) property, such that the concatenation \(f=f_1||f_2||f_3||f_4\) does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations \(\pi _i\) 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; 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.