Constructing and Counting Even-Variable Symmetric Boolean Functions with Algebraic Immunity not Less Than $d$

TL;DR

This paper presents a method to explicitly construct a large class of symmetric Boolean functions with high algebraic immunity, including maximum immunity, and establishes a new lower bound for such functions.

Contribution

It introduces a novel construction of symmetric Boolean functions with guaranteed algebraic immunity and derives the first known lower bound for their quantity.

Findings

Constructed 2^{loor{ ext{log}_2{k}}+2} functions with maximum algebraic immunity.

Provided a new lower bound 2^{loor{ ext{log}_2{d}}+2(k-d+1)} for functions with immunity at least d.

Achieved a significantly larger set of functions than previous methods.

Abstract

In this paper, we explicitly construct a large class of symmetric Boolean functions on $2k$ variables with algebraic immunity not less than $d$, where integer $k$ is given arbitrarily and $d$ is a given suffix of $k$ in binary representation. If let $d = k$, our constructed functions achieve the maximum algebraic immunity. Remarkably, $2^{\lfloor \log_2{k} \rfloor + 2}$ symmetric Boolean functions on $2k$ variables with maximum algebraic immunity are constructed, which is much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than $d$ is derived, which is $2^{\lfloor \log_2{d} \rfloor + 2(k-d+1)}$. As far as we know, this is the first lower bound of this kind.