Monomial Boolean functions with large high-order nonlinearities

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 monomial Boolean functions.

We prove lower bounds on the second-order nonlinearities of functions ${\mathrm{tr}}_n\left(x^7\right)$ and ${\mathrm{tr}}_n\left(x^{2^r+3}\right)$ where $n=2r$. Among all monomial Boolean functions, our bounds match the best second-order nonlinearity lower bounds by Carlet [IEEE Transactions on Information Theory 54(3), 2008] and Yan and Tang [Discrete Mathematics 343(5), 2020] for odd and even n, respectively. We prove a lower bound on the third-order nonlinearity for functions ${\mathrm{tr}}_n\left(x^{15}\right)$, which is the best third-order nonlinearity lower bound. For any r, we prove that the r-th order nonlinearity of ${\mathrm{tr}}_n\left(x^{2^{r+1}-1}\right)$ is at least $2^{n-1}-2^{(1-2^{-r})n+\frac{r}{2^{r-1}}-1}-O\left(2^{\frac{n}{2}}\right)$. For $r\ll {\log }_{2}n$, this is the best lower bound among all explicit functions.