Bent and bent4 spectra of Boolean functions over finite fields

For c∈F2n, a c-bent4 function f from the finite field F2n to F2 is a function with a flat spectrum with respect to the unitary transform Vfc, which is designed to describe the component functions of modified planar functions. For c=0 the transform Vfc reduces to the conventional Walsh transform, and hence a 0-bent4 function is bent. In this article we generalize the concept of partially bent functions to the transforms Vfc. We show that every quadratic function is partially bent, and hence it is plateaued with respect to any of the transforms Vfc. In detail we analyse two quadratic monomials. The first has values as small as possible in its spectra with respect to all transforms Vfc, and the second has a flat spectrum for a large number of c. Moreover, we show that every quadratic function is c-bent4 for at least three distinct c. In the last part we analyse a cubic monomial. We show that it is c-bent4 only for c=1, the function is then called negabent, which shows that non-quadratic functions exhibit a different behaviour.