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 mathbbF2n{mathbb{F}_{2^{n}}} (n = 2m) having the form f(x) = tro(s1) (a xs1) + tro(s2) (b xs2){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 F2o(s1){F_{2^{o(s_1)}}} and F2o(s2){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 s2=frac 2n-13{s_2={frac {2^n-1}3}}, where a Î mathbbF2n{ainmathbb{F}_{2^{n}}} (a ≠ 0) and b Î mathbbF4{binmathbb{F}_{4}} provide a construction of bent functions over mathbbF2n{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.
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机译：Bent函数是最大程度的非线性布尔函数，仅对于输入数为偶数的函数存在。本文对在具有以下特征的mathbbF 2 n {mathbb {F} _ {2 ^ {n}}}（n = 2m）上构造弯曲函数做出了贡献形式f（x）= tr o（s 1 ）（ax s 1 ）+ tr o（s 2 ）（bx s 2 ）{f（x）= tr_ {o（s_1）}（ax ^ {s_1}）+ tr_ {o（s_2）}（bx ^ {s_2}）}其中o（s i ）表示2模2 n < / sup> − 1包含s i ，系数a和b分别在F 2 o（s 1 ）中 {F_ {2 ^ {o（s_1）}}}和F 2 o（s 2 ） {F_ {2 ^ {o（s_2）}}}。文献中介绍了多项式弯曲函数的许多构造，但即使在二项式情况下也鲜为人知。我们证明了指数s 1 = 2 m − 1和s 2 = frac 2 n -13 { s_2 = {frac {2 ^ n-1} 3}}，其中ÎmathbbF 2 n {ainmathbb {F} _ {2 ^ {n}}}}（ a≠0）和bÎmathbbF 4 {binmathbb {F} _ {4}}在mathbbF 2 n 上提供了一个弯曲函数的构造。 {mathbb {F} _ {2 ^ {n}}}，具有最佳代数度。奇怪的是，我们用Kloosterman之和来明确描述这些函数的弯曲性。我们将指数s 1 的形式为r（2 m − 1）的函数推广化，其中r与2 m + 1.相应的弯曲功能也超弯曲。甚至对于m，我们根据这些Kloosterman之和给出弯曲的必要条件。
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