A Note on Semi-bent and Hyper-bent Boolean Functions

Abstract

Semi-bent and hyper-bent funcitons as two classes of Boolean functions with low Walsh transform, are applied in cryptography and commnunications. This paper considers a new class of semi-bent quadratic Boolean function and a generalization of a new class of hyper-bent Boolean functions. The new class of semi-bent quadratic Boolean function of the form \(f(x)=\sum _{i=1}^{\lfloor \frac{m-1}{2}\rfloor }Tr^n_1(c_ix^{1+4^{i}}) (c_i\in \mathbb {F}_4\),\(n=2m)\) is simply characterized and enumerated. Then we present the characterization of a generalization of a new class of hyper-bent Boolean functions of the form \(f^{(r)}_{a,b}:=\mathrm {Tr}_{1}^{n}(ax^{r(2^m-1)}) +\mathrm {Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})\), where \(n=2m\), \(m\equiv 2\pmod 4\), \(a\in \mathbb {F}_{2^m}\) and \(b\in \mathbb {F}_{16}\).