**FURTHER PROPERTIES OF SEVERAL CLASSES OF BOOLEAN FUNCTIONS WITH OPTIMUM ALGEBRAIC IMMUNITY**

*Claude Carlet and Xiangyong Zeng and Chunlei Li and Lei Hu*

**Abstract: **Thanks to a method proposed by Carlet, several classes of balanced
Boolean functions with optimum algebraic immunity are obtained. By
choosing suitable parameters, for even $n\geq 8$, the balanced
$n$-variable functions can have nonlinearity
$2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}/(n-2)$,
and for odd $n$, the functions can have nonlinearity
$2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)$, where the function
$\Delta(n)$ is describled in Theorem 4.4. The algebraic
degree of some constructed functions is also discussed.

**Category / Keywords: **foundations / boolean functions

**Date: **received 14 Sep 2007

**Contact author: **xzeng at hubu edu cn

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

**Version: **20070919:211553 (All versions of this report)

**Short URL: **ia.cr/2007/370

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