Modified planar functions and their components

Zhou ([20]) introduced modified planar functions in order to describe (2ⁿ; 2ⁿ; 2ⁿ; 1) relative difference sets R as a graph of a function on the finite field F_2n, and pointed out that projections of R are difference sets that can be described by negabent or bent₄ functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. Moreover, we obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function. We finally point out that though these components behave somewhat different than the multivariate bent4 functions, they are bent or semibent functions shifted by a certain quadratic term, a property which they share with their multivariate counterpart.