## Cryptology ePrint Archive: Report 2012/335

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