On Construction of Perfect Ciphers

K. Shannon in the 40s of the 20th century introduced the concept of a perfect cipher, which provides the best protection of plaintexts. Perfect secrecy means that cryptanalyst can obtain no information about the plaintext by observing the ciphertext. In the paper we study the problem of construction of perfect ciphers on a given set of plaintexts $X$, a set of keys $K$ and a probability distribution $P(K)$. We give necessary and sufficient conditions for a perfect ciphers on given $X$, $K$ and $P(K)$. It is shown that this problem is reduced to construction of the set of partitions of the set $K$ with certain conditions. As one of the drawbacks of the probability model of cipher are limitations on the power of sets of plaintexts, keys and ciphertexts we also study the problem of construction of substitution cipher with unbounded key on a given set of ciphervalues, a set of keys and a probability distribution on the set of keys.