A Note on “On the Construction of Boolean Functions with Optimal Algebraic Immunity”

Abstract

In this note, we go further on the “basis exchange” idea presented in [2] by using Mobious inversion. We show that the matrix S₁(ƒ)S₀(ƒ)^-1 has a nice form when ƒ is chosen to be the majority function, where S₁(ƒ) is the matrix with row vectors υ_k(α) for all α ∈ 1_ƒ and S₀(ƒ)=S₁(ƒ ⊕ 1). And an exact counting for Boolean functions with maximum algebraic immunity by exchanging one point in on-set with one point in off-set of the majority function is given. Furthermore, we present a necessary condition according to weight distribution for Boolean functions to achieve algebraic immunity not less than a given number.