Homophonic substitution

Abstract

Historically most of the secret-key cryptographic systems that have been broken were broken by exploiting the deviation of the statistics of the clear-text from that of a completely random sequence. Homophonic substitution is a venerable technique for converting a clear-text sequence into a random sequence. In 1988 Günther introduced an important generalization of homophonic substitution called variable-length homophonic substitution.The purpose of this presentation is first to review the information-theoretic treatment of Günther's homophonic substitution and then show how to implement it with a finite memory, considering clear-text symbol probabilities which are rational numbers. Shannon's concept of a strongly-ideal cipher system will be reviewed in order to provide the motivation for using any kind of homophonic substitution. The precise definition of variable-length homophonic substitution is presented together with the necessary and sufficient condition for such a substitution to be perfect, i.e., to create a completely random sequence. By employing binary coding, perfect homophonic substitution can be achieved with the introduction of less than 2 bits of entropy in each encoded source letter, and can be implemented using less than 4 random bits per coded letter. Some properties of the geometric series, resulting from the base 2 expansion of the clear-text symbol probabilities, are presented and are used to establish an accurate lower bound for the redundancy in homophonic substitution. - ( May 24, 2002 ) .

The entropy of a code with probabilities

“…The proof of Shannon's noiseless source coding theorem [8, p.691 implicitly suggests that the only way to reduce the redundancy of a message to be transmitted or stored is by encoding extensions of the original source. In [9] we introduced an iterative procedure to reduce redundancy, as an alternative to coding the extensions of a source. We remark…”