A new class of $ p $ -ary regular bent functions
Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou · Advances in Mathematics of Communications · 2019
Bent functions have many important applications in cryptography and coding theory. This paper considers a class of \begin{document}$ p $\end{document} -ary functions with the Dillon exponent of the form \begin{document}$ f(x) = \sum\limits_{i = 0}^{q-1}(Tr^n_1(a_1x^{(r i+s)(q-1)})+Tr^n_1(a_2x^{(r i+s)(q-1)+\frac{q^2-1}{2}}))+bx^{\frac{q^2-1}{2}}, $\end{document} where \begin{document}$ n = 2m $\end{document} , \begin{document}$ q = p^m $\end{document} , \begin{document}$ p $\end{document} is an odd prime, \begin{document}$ a_1,a_2\in \mathbb{F}_{p^n} $\end{document} , and \begin{document}$ b\in \mathbb{F}_p $\end{document} . With the help of Kloosterman sums, we present an explicit characterization of these \begin{document}$ p $\end{document} -ary regular bent functions for the case \begin{document}$ gcd(s-r,\frac{q+1}{2}) = 1 $\end{document} and \begin{document}$ gcd(r,q+1) = 1 $\end{document} or \begin{document}$ 2 $\end{document} . Our results generalize results of Li et al. [IEEE Trans. Inf. Theory 59 (2013) 1818-1831].