A Note on the Optimal Immunity of Boolean Functions Against Fast Algebraic Attacks

  • Jing Shen
  • Yusong DuEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)


The immunity of Boolean functions against fast algebraic attacks is an important cryptographic property. When deciding the optimal immunity of an n-variable Boolean function against fast algebraic attacks, one may need to compute the ranks of a series of matrices of size \(\sum _{i=d+1}^{n}{n \atopwithdelims ()i}\times \sum _{i=0}^e{n \atopwithdelims ()i}\) over binary field \(\mathbb {F}_2\) for each positive integer e less than \(\lceil \frac{n}{2}\rceil \) and corresponding d. In this paper, for an n-variable balanced Boolean function, exploiting the combinatorial properties of the binomial coefficients, when n is odd, we show that the optimal immunity is only determined by the ranks of those matrices such that \(\sum _{i=0}^e{n \atopwithdelims ()i}\) is even. When n is even but not the power of 2, we show that the optimal immunity is only determined by the ranks of those matrices such that \(\sum _{i=0}^e{n \atopwithdelims ()i}\) is even or such that both \(\sum _{i=0}^e{n \atopwithdelims ()i}\) and \(\sum _{i=0}^{e+1}{n \atopwithdelims ()i}\) are odd.


Boolean function Fast algebraic attack Algebraic immunity 


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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Guangdong College of Industry and CommerceGuangzhouChina
  2. 2.School of Information ManagementSun Yat-sen UniversityGuangzhouChina

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