A New Family of Hyper-Bent Boolean Functions in Polynomial Form

Abstract

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. 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. (Hyper)-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. Few constructions of hyper-bent functions defined over the Galois field ${\mathbb F}_{2n}$ (n = 2m) are proposed in the literature. The known ones are mostly monomial functions.

This paper is devoted to the construction of hyper-bent functions. We exhibit an infinite class over ${\mathbb F}_{2n}$ (n = 2m, m odd) having the form $f(x) = Tr_1^{o(s_1)} (a x^{s_1}) + Tr_1^{o(s_2)} (b x^{s_2})$ where o(s _i ) denotes the cardinality of the cyclotomic class of 2 modulo 2ⁿ − 1 which contains s _i and whose coefficients a and b are, respectively in ${\mathbb F}_{2^{o(s_1)}}$ and ${\mathbb F}_{2^{o(s_2)}}$. We prove that the exponents $s_1={3(2^m-1)}$ and $s_2={\frac {2^n-1}3}$, where $a\in {\mathbb F}_{2n}$ ($a\not=0$) and $b\in{\mathbb F}_4$ provide a construction of hyper-bent functions over ${\mathbb F}_{2n}$ with optimum algebraic degree. We give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums and the cubic sums involving only the coefficient a.