Cryptanalysis of an asymmetric cipher on Boolean functions

The asymmetric encryption system ACBF is considered. Its key is an invertible vectorial Boolean function constructing from a generating function (which is considered known) using the negation and permutation operations of variables and coordinates. Of these four operations, some are identical, the rest form a set of key parameters; finding them is the goal of the attack. For seven of 15 possible sets of key parameters, attacks with known plaintext are described, their complexity is given. For five sets of key parameters, attacks with choosen plaintext are presented too. The main stage of the attacks is the solution of the auxiliary problem of finding a columns permutation, with the means of which one Boolean matrix is obtained from another. It has been proved that, for uniquely determing the key, it is necessary to have $2\log n$ plaintexts (in average) in the attack with a known plaintext, and it is enough to choose $\log n$ plaintexts in the attack with a choosen plaintext, where $n$ is the length of text.