  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.
