The criterion of minimum perfect ciphers with respect to inclusion

The paper deals with the problem of Shannon perfect ciphers description (which are absolutely immune against the attack on ciphertext, according to Shannon), minimal by inclusion. The criterion of minimum non-endomorphic (endomorphic) perfect ciphers by inclusion is formulated and proved. The table of encryption of a perfect cipher with $\lambda>1$ ciphers, $\mu\geq\lambda$ ciphers and $\pi\geq\mu$ keys is considered. For a given cipher, $(0,1)$-matrix with $\pi$ rows and $1+\lambda\mu$ columns is constructed in a natural way. It is shown that the set of encryption keys is minimal if and only if the matrix rank is maximal and equals to $\pi$. The necessary conditions for perfect ciphers of minimum by inclusion have been obtained.