Construction of balanced even-variable Boolean functions with optimal algebraic immunity

Abstract

Given a positive integer n, let $k=\lceil n/2\rceil -1$ and $s=\sum _{i=0}^k\left(\begin{array}{c}n\\ i\end{array}\right)$. The generator matrix $G(k,n{)}_{s\times 2^n}$ of the kth-order Reed–Muller code RM$(k,n)$ is an important tool in the study of Boolean functions' algebraic immunity. In this paper, choosing the last s column vectors in $G(k,n)$ as a basis of the vector space ${\mathbb{F}}_2^s$, we study the values of the coefficients in the linear expressions of $G(k,n)$'s column vectors over this basis. As an application, we present a new construction of balanced even-variable Boolean functions on ${\mathbb{F}}_2^n$ with optimal algebraic immunity by modifying the outputs of majority function. The nonlinearities of these constructed functions are also determined.

Keywords:

• Boolean function
• Reed–Muller code
• algebraic immunity
• Walsh transform
• nonlinearity

2000 AMS Subject Classifications:

• 94A60
• 68P25

Acknowledgments

The author thank the two anonymous referees and the editor for their helpful suggestions and comments that improved the quality of this paper and they are indebted to Simon Fischer for the evaluation of the resistance of the functions to fast algebraic attacks. This work was supported by the National Natural Science Foundation of China [grant number 61201253].