Cryptology: The Mathematics of Secure Communication

Conclusion

If the model developed here for the abstract encryption/decryption channel is as general as is claimed, the discovery of asymmetric encryption techniques may be the ultimate revolution in cryptography. However, even if this should prove to be true, the impact on the practice of cryptography will continue for a very long time. For example, the mechanization of the encryption/decryption functions using computing elements, which began almost fifty years ago, has in just the past year progressed to a point where the NBS data encryption standard (DES) — a symmetric encryption scheme with a 64 bit key space [36,37] — is now offered on a single LSI chip by three manufacturers, and a two-chip MOS realization of the M.I.T. scheme with an eighty-decimal modulus has been designed. Since, in this article we were concerned more with the theory of secure communications than with the practice, no mention was made of the very significant fact that all of the asymmetric schemes which have been proposed thus far exact an extremely high price for their asymmetry — the increased amount of computation required in the encryption/decryption process cuts the channel capacity (bits per second of message information communicated) dramatically. In fact, at the moment no asymmetric scheme (to the best of the author’s knowledge) has been able to break theC ^1/2 bound, whereC is the channel capacity of a symmetric channel having the same cryptosecurity and using the same basic clock or bit manipulation rate. If this difference is genuine, as we believe it to be, and not just an artifact of the asymmetric schemes which happen to have been considered, then both symmetric and asymmetric encryption/decryption schemes will be needed depending on the requirements of each application — and asymmetric techniques will not supplant symmetric techniques in general.

We said earlier that the investigation of the abstract encryption/decryption channel is the most important question in contemporary applied mathematics; others have characterized it as a multimillion dollar problem [38] awaiting solution. Irrespective of the accuracy of these judgements though, we would hope that a compelling case has been presented that contemporary cryptology is an exciting and important mathematical discipline opening challenging new problems in many areas of mathematics.