Refinements and Extensions of Ziv's Model of Perfect Secrecy for Individual Sequences

We refine and extend Ziv's model and results regarding perfectly secure encryption of individual sequences. According to this model, the encrypter and the legitimate decrypter share in common a secret key, not shared with the unauthorized eavesdropper, who is aware of the encryption scheme and has some prior knowledge concerning the individual plaintext source sequence. This prior knowledge, combined with the cryptogram, is harnessed by eavesdropper which implements a finite-state machine as a mechanism for accepting or rejecting attempted guesses of the source plaintext. The encryption is considered perfectly secure if the cryptogram does not provide any new information to the eavesdropper that may enhance its knowledge concerning the plaintext beyond his prior knowledge. Ziv has shown that the key rate needed for perfect secrecy is essentially lower bounded by the finite-state compressibility of the plaintext sequence, a bound which is clearly asymptotically attained by Lempel-Ziv compression followed by one-time pad encryption. In this work, we consider some more general classes of finite-state eavesdroppers and derive the respective lower bounds on the key rates needed for perfect secrecy. These bounds are tighter and more refined than Ziv's bound and they are attained by encryption schemes that are based on different universal lossless compression schemes. We also extend our findings to the case where side information is available to the eavesdropper and the legitimate decrypter, but may or may not be available to the encrypter as well.