On $2k$-Variable Symmetric Boolean Functions with Maximum Algebraic Immunity $k$

TL;DR

This paper characterizes the weight distribution of symmetric Boolean functions with maximum algebraic immunity and provides a construction method for all such functions based on the binary expansion of the number of variables.

Contribution

It determines the weight distribution of symmetric Boolean functions with maximum algebraic immunity and constructs all such functions explicitly.

Findings

Weight distribution depends on the binary expansion of n.

Explicit construction of all symmetric Boolean functions with maximum algebraic immunity.

The number of such functions is given by a formula involving the binary weight of n.

Abstract

Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean function with maximum algebraic immunity $\frac{n}{2}$ is determined by the binary expansion of $n$. Based on the foregoing, all $n$-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is $(2\wt(n)+1)2^{\lfloor \log_2 n \rfloor}$