A new class of bent and hyper-bent Boolean functions in polynomial forms

Abstract

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ (n = 2m) having the form $${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$$ where o(s i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 n ? 1 which contains s i and whose coefficients a and b are, respectively in $${F_{2^{o(s_1)}}}$$ and $${F_{2^{o(s_2)}}}$$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s 1 = 2 m ? 1 and $${s_2={\frac {2^n-1}3}}$$ , where $${a\in\mathbb{F}_{2^{n}}}$$ (a ? 0) and $${b\in\mathbb{F}_{4}}$$ provide a construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s 1 is of the form r(2 m ? 1) where r is co-prime with 2 m + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.