The characterization of hyper-bent function with multiple trace terms in the extension field

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~F_2²ᵐ~of this field~F_2ᵐ. By applying M\"{o}bius transformation and the theorems of hyperelliptic curves, hyper-bentness of these functions are successfully characterized in this field~F_2²ᵐ with~m~odd integer.