Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order nonlinearities of some trace monomial Boolean functions. We prove lower bounds on the second-order nonlinearities of functions $\mathrm{tr}_n(x^7)$ and $\mathrm{tr}_n(x^{2^r+3})$ where $n=2r$. Among all trace monomials, our bounds match the best second-order nonlinearity lower bounds by \cite{Car08} and \cite{YT20} for odd and even $n$ respectively. We prove a lower bound on the third-order nonlinearity for functions $\mathrm{tr}_n(x^{15})$, which is the best third-order nonlinearity lower bound. For any $r$, we prove that the $r$-th order nonlinearity of $\mathrm{tr}_n(x^{2^{r+1}-1})$ is at least $2^{n-1}-2^{(1-2^{-r})n+\frac{r}{2^{r-1}}-1}- O(2^{\frac{n}{2}})$. For $r \ll \log_2 n$, this is the best lower bound among all explicit functions.

翻译：在密码学、编码理论和计算复杂性中，展示一个具有大高阶非线性度的显式布尔函数是一个重要问题。我们证明了某些迹单项布尔函数的二阶、三阶及更高阶非线性度的下界。对于函数 $\mathrm{tr}_n(x^7)$ 和 $\mathrm{tr}_n(x^{2^r+3})$，其中 $n=2r$，我们证明了其二阶非线性度的下界。在所有迹单项中，我们的下界分别与文献 \cite{Car08} 和 \cite{YT20} 在奇数 $n$ 和偶数 $n$ 情况下最优的二阶非线性度下界一致。对于函数 $\mathrm{tr}_n(x^{15})$，我们证明了其三阶非线性度的下界，这是最优的三阶非线性度下界。对于任意 $r$，我们证明 $\mathrm{tr}_n(x^{2^{r+1}-1})$ 的 $r$ 阶非线性度至少为 $2^{n-1}-2^{(1-2^{-r})n+\frac{r}{2^{r-1}}-1}- O(2^{\frac{n}{2}})$。当 $r \ll \log_2 n$ 时，这是所有显式函数中最优的下界。