The characterization of hyper-bent function with multiple trace terms in the extension field
Peng Han, Keli Pu
TL;DR
The work addresses the classification of hyper-bent functions with multiple trace terms and Dillon-like exponents over the extension field $\mathbb{F}_{2^{2m}}$ (with $m$ odd). It develops a framework based on Möbius transformations and hyperelliptic-curve theory to characterize hyper-bentness via transformed exponential sums and algebraic-geometry counts, culminating in a set of criteria (Theorems 3.1–3.6) that connect $\Lambda(f)$, trace polynomials $g_{a'_{r}}$, and the curve counts $\#\mathcal{C}(\mathbb{F}_{2^{m}})$. These results extend Charpin-Gong–type families to coefficient sets in the full extension field and resolve a standing open problem by providing explicit, equivalent conditions for hyper-bentness, with potential cryptographic implications. The work also suggests avenues for generalizing to other trace-term configurations and to $p$-ary hyper-bent settings, highlighting the deep links between finite-field transforms, Dickson polynomials, and hyperelliptic curves in function classification.
Abstract
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 Möbius 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.