The lower bounds on the second-order nonlinearity of three classes of Boolean functions

In this paper, by calculating the lower bounds on the nonlinearity of the derivatives of the following three classes of Boolean functions, we provide the tight lower bounds on the second-order nonlinearity of these Boolean functions: (1) \begin{document}$ f_1(x) = Tr_1^n(x^{2^{r+1}+2^r+1}) $\end{document} , where \begin{document}$ n = 2r+2 $\end{document} with even \begin{document}$ r $\end{document} ; (2) \begin{document}$ f_2(x) = Tr_1^n(\lambda x^{2^{2r}+2^{r+1}+1}) $\end{document} , where \begin{document}$ \lambda \in \mathbb{F}_{2^r}^* $\end{document} and \begin{document}$ n = 4r $\end{document} with even \begin{document}$ r $\end{document} ; (3) \begin{document}$ f_3(x,y) = yTr_1^n(x^{2^r+1})+Tr_1^n(x^{2^r+3}) $\end{document} , where \begin{document}$ (x, y)\in \mathbb{F}_{2^n}\times \mathbb{F}_2 $\end{document} , \begin{document}$ n = 2r $\end{document} with odd \begin{document}$ r $\end{document} . The results show that our bounds are better than previously known lower bounds in some cases.