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

Abstract

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.

This work is supported by National Natural Science Foundations of China (Grant No. 61309028, Grant No. 61472457, Grant No. 61502113), Science and Technology Planning Project of Guangdong Province, China (Grant No. 2014A010103017), and Natural Science Foundation of Guangdong Province, China (Grant No. 2016A030313298).