**Third-order nonlinearities of some biquadratic monomial Boolean functions**

*Brajesh Kumar Singh*

**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}

**Category / Keywords: **Boolean functions, Walsh-Hadamard spectrum, Third-order nonlinearities, Linearized polynomial

**Date: **received 8 Apr 2012, last revised 8 Apr 2012

**Contact author: **bksingh0584 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Note: **Dear sir,

Some typos errors are removed here.

Thank you.

**Version: **20120411:204848 (All versions of this report)

**Short URL: **ia.cr/2012/186

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