Some generalizations of Shannon's theory of perfect ciphers

K. Shannon in the 1940s introduced the concept of a perfect cipher, which provides the best protection of plaintexts. Perfect secrecy means that a cryptanalyst can obtain no information about the plaintext by observing the ciphertext. It is well known that the Vernam cipher with equiprobable gamma is a perfect cipher but it is not imitation resistant because it uses equipotent alphabets for plaintexts and ciphertexts. Also in this cipher should be used equiprobable key sequences that are not always reached. In this review paper discusses the problems of constructing perfect and $(k|y)$-perfect ciphers for a given set of parameters. We give necessary and sufficient conditions for these ciphers. We construct perfect and $(k|y)$-perfect substitution ciphers with unlimited key and imitation resistant perfect ciphers. We study the case when the random generator of key sequences does not necessarily have a uniform probability distribution.