Constructing vectorial Boolean functions with high algebraic immunity based on group decomposition

Abstract

In this paper, we propose a new method to construct cryptographically significant (vectorial) Boolean functions over by using the decomposition of the multiplicative group of , and give a generalized description for constructing a class of vectorial Boolean functions with high algebraic immunity. Moreover, for even n, we introduce two special classes of vectorial Boolean functions with high algebraic immunity: one is hyper-bent and the other is balanced with optimal algebraic degree.

Keywords:

• vectorial Boolean function
• polar decomposition
• algebraic immunity
• balancedness
• algebraic degree
• hyper-bent function

2010 AMS Subject Classifications:

• 06E30
• 11L05

Acknowledgements

The authors thank the reviewers for their helpful advice and comments. This work is supported by the Science and Technology on Information Assurance Laboratory under Grant No.KJ-11-02 and National Natural Science Foundation of China under Grants No. 61272499, No. 10990011.