Cryptanalysis of the ACBF encryption system

The ACBF encryption system with a functional key is considered. A public key in the cryptosystem is a vectorial Boolean function $f$ in $n$ variables obtained by permutation and negation operations on variables and coordinate functions of a bijective vectorial Boolean function $g$, that is, $f(x)=\pi_2(g^{\sigma_2}(\pi_1(x^{\sigma_1})))$, $\pi_1,\pi_2\in\mathbb{S}_n$ and $\sigma_1,\sigma_2\in\mathbb{F}_2^n$ are key parameters. A private key is $f^{-1}$. For two subsets of key parameters, namely for $\{\pi_1\}$ and $\{\pi_1,\pi_2\}$, attacks with known plaintexts are proposed.