Constructions of 2-rotation symmetric bent functions based on Maiorana–McFarland’s bent function

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for more than five decades. Rotation symmetric Boolean functions, introduced by Pieprzyk and Qu, are those Boolean functions which are invariant under the cyclic shifts of inputs. In this paper, we first give two flexible construction methods of bent functions on n variables by defining two subsets T ’s of F 2 n to modify the support of Maiorana–McFarland’s bent function f 0 ( x , y ) = x ⋅ π ( y ) ⊕ h ( y ) , where n = 2 m , x , y ∈ F 2 m , π is a permutation over F 2 m , and h is a Boolean function on m variables. These methods have corresponding constraints on π and h . Then, we deduce the dual functions of the newly constructed bent functions. Lastly, we propose the methods of constructing 2-rotation symmetric bent functions by redefining the two subsets T ’s of F 2 n and by imposing more restrictive constraints on π and h .