Construction of bent functions via Niho power functions

A Boolean function with an even number n = 2k 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(α1x1 + α2x2), α1, α2, x ∈ F2n , are considered, where the exponents di (i = 1, 2) are of Niho type, i.e. the restriction of xi on F2k is linear. We prove for d1 = 2 + 1 and d2 = 3 · 2k−1 − 1, d2 = 2 + 3 if k is odd, d2 = (2 + 5)/3 if k is even, resp., that f is a bent function if α1 + α1 = 1 and α2 = 1.