Construction of bent functions via Niho power functions

A Boolean function with an even number n = 2 k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f ( x ) = tr ( α 1 x d 1 + α 2 x d 2 ) , α 1 , α 2 , x ∈ F 2 n , are considered, where the exponents d i ( i = 1 , 2 ) are of Niho type, i.e. the restriction of x d i on F 2 k is linear. We prove for several pairs of ( d 1 , d 2 ) that f is a bent function, when α 1 and α 2 fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F 2 n are bijective.