Hyper-bent Boolean Functions with Multiple Trace Terms

Abstract

Bent functions are maximally nonlinear Boolean functions with an even number of variables. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. In fact, hyper-bent functions seem still more difficult to generate at random than bent functions and many problems related to the class of hyper-bent functions remain open. (Hyper)-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. In this paper, we contribute to the knowledge of the class of hyper-bent functions on finite fields F2n (where n is even) by studying a subclass Fn of the so-called Partial Spreads class PS- (such functions are not yet classified, even in the monomial case). Functions of Fn have a general form with multiple trace terms. We describe the hyper-bent functions of Fn and we show that the bentness of those functions is related to the Dickson polynomials. In particular, the link between the Dillon monomial hyper-bent functions of Fn and the zeros of some Kloosterman sums has been generalized to a link between hyper-bent functions of Fn and some exponential sums where Dickson polynomials are involved. Moreover, we provide a possibly new infinite family of hyper-bent functions. Our study extends recent works of the author and is a complement of a recent work of Charpin and Gong on this topic.