Bent functions are maximally nonlinear Boolean functions with an even number of variables, which include a subclass of functions, the so-called hyper-bent functions whose properties are stronger than bent functions and a complete classification of hyper-bent functions is elusive and inavailable.~In this paper,~we solve an open problem of Mesnager that describes hyper-bentness of hyper-bent functions with multiple trace terms via Dillon-like exponents with coefficients in the extension field~$\mathbb{F}_{2^{2m}}$~of this field~$\mathbb{F}_{2^{m}}$. By applying M\"{o}bius transformation and the theorems of hyperelliptic curves, hyper-bentness of these functions are successfully characterized in this field~$\mathbb{F}_{2^{2m}}$ with~$m$~odd integer.

翻译：弯曲函数是变量个数为偶数的极大非线性布尔函数，其中包含一个函数子类，即所谓的超弯曲函数，其性质强于弯曲函数，且超弯曲函数的完整分类尚不明确且难以获得。~本文中，~我们解决了Mesnager提出的一个公开问题，该问题通过具有Dillon类指数且系数在扩域~$\mathbb{F}_{2^{2m}}$~（该域为~$\mathbb{F}_{2^{m}}$~的扩域）中的多迹项来描述超弯曲函数的超弯曲性。通过应用Möbius变换和超椭圆曲线定理，我们成功地在~$m$~为奇数的域~$\mathbb{F}_{2^{2m}}$~中刻画了这些函数的超弯曲性。