Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials

Abstract

This paper is devoted to the study of hyperbent functions in n variables, i.e., bent functions which are bent up to a change of primitive roots in the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). Our main purpose is to obtain an explicit trace representation for some classes of hyperbent functions. We first exhibit an infinite class of monomial functions which is not hyperbent. This result indicates that Kloosterman sums on F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> cannot be zero at some points. For functions with multiple trace terms, we express their spectra by means of Dickson polynomials. We then introduce a new tool to describe these hyperbent functions. The effectiveness of this new method can be seen from the characterization of a new class of binomial hyperbent functions.