**Constructing Vectorial Boolean Functions with High Algebraic Immunity Based on Group Decomposition**

*Yu Lou and Huiting Han and Chunming Tang and Maozhi Xu*

**Abstract: **In this paper, we construct a class of vectorial Boolean functions over $\mathbb{F}_{2^{n}}$ with high algebraic immunity based on the decomposition of the multiplicative group of $\mathbb{F}_{2^n}$. By viewing $\mathbb{F}_{2^{n}}$ as $G_1G_2\bigcup \{0\} $ (where $G_1$ and $G_2$ are subgroups of $\mathbb{F}_{2^{n}}^{*},~(\#G_1,\#G_2)=1$ and $\#G_1\times \#G_2=2^{2k}-1$), we give a generalized description for constructing vectorial Boolean functions with high algebraic immunity. Moreover, when $n$ is even, we provide two special classes of vectorial Boolean functions with high(sometimes optimal) algebraic immunity, one is hyper-bent, and the other is of balancedness and optimal algebraic degree .

**Category / Keywords: **foundations / vectorial Boolean function, polar decomposition, algebraic immunity, balancedness, algebraic degree, hyper-bent functions

**Date: **received 12 Jun 2012

**Contact author: **windtker at pku edu cn

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

**Version: **20120622:193310 (All versions of this report)

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