A generalization of hellman’s extension of shannon’s approach to cryptography

Abstract

In his landmark 1977 paper [Hell77], Hellman extends the Shannon theory approach to cryptography [Shan49]. In particular, he shows that the expected number of spurious key decipherements on length n messages is at least 2H(K)-nD - 1 for any uniquely enci- pherable, uniquely decipherable cipher, as long as each key is equally likely and the set of meaningful cleartext messages follows a uniform distribution (where H(K) is the key entropy and D is the redundancy of the source language). In this paper, we show that Hellman’s result holds with no restrictions on the distribution of keys and messages. We also bound from above and below the key equivocation upon seeing the ciphertext. Sufficient conditions for these bounds to be tight are given. The results are obtained through very simple purely information theoretic arguments, with no needs for (explicit) counting arguments.