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arXiv · 1110.3876

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

Yuan Li·Haibin Kan·Futatsugi Kokichi
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Abstract

In this note, we go further on the "basis exchange" idea presented in \cite{LiNa1} by using Mobious inversion. We show that the matrix $S_1(f)S_0(f)^{-1}$ has a nice form when $f$ is chosen to be the majority function, where $S_1(f)$ is the matrix with row vectors $\upsilon_k(\alpha)$ for all $\alpha \in 1_f$ and $S_0(f)=S_1(f\oplus1)$. 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.

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Yuan Li, Haibin Kan, Futatsugi Kokichi. 2011-10-18. A Note on "On the Construction of Boolean Functions with Optimal Algebraic Immunity". https://doi.org/10.1587/transfun.e94.a.1877

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