Information theory and public key cryptosystems

Abstract

Abstract : Shannon has defined the unicity distance of a random cipher as the point where there is no uncertainty over which key was used for enciphering. The unicity distance is given as a value N where N = cryptogram length in characters. The usual issue for classical cryptography is: given ciphertext (and possibly corresponding plaintext) under the assumption of a random cipher, is this information sufficient on the average to determine the key. Here, if we let M denote the random variable (defined as the number of keys that will decipher a given intercepted cryptogram into a meaningful message), it turns out that M has a binomial distribution. Meyer and Matyas have expanded Shannon's approach to unicity distance by using information theory. They make no assumption about the distribution of M so their approach applies to cryptosystems in general. This paper applies this method to public key cryptography. In particular, we consider the RSA (Rivest-Shamir-Adelman) cryptosystem which is probably the most widely known public key system. The motivation for this work was to investigate the complexity of RSA. No new research has been conducted into the information theoretic approach to cryptosystem security since Shannon's work. This report shows that the present state of this theory is inadequate to handle public key cryptosystems like the RSA system. It is hoped that with further research on this topic (e.g., with making proper assumptions), the information theoretic approach can be made to at least give some kind of theoretical bound for the security of public key systems.