Third-order nonlinearities of some biquadratic monomial Boolean functions

Abstract

In this paper, we estimate the lower bounds on third-order nonlinearities of some biquadratic monomial Boolean functions of the form $Tr_1^n(\lambda x^d)$ for all $x \in \mathbb F_{2^n}$, where \lambda \in \BBF_{2^n}^{*}, \begin{itemize} \item [{(1)}]$d = 2^i + 2^j + 2^k + 1$, $i, j, k$ are integers such that $i > j > k \geq 1$ and $n > 2 i$. \item [{(2)}] $d = 2^{3\ell} + 2^{2\ell} + 2^{\ell} + 1$, $\ell$ is a positive integer such that $\gcd (i, n) = 1$ and $n > 6$. \end{itemize