Discrete Logarithms and Elliptic Curves in Cryptography

Since ancient times, there has been a tug-of-war taking place between code makers and code breakers. Only within the last fifty years have the code makers emerged victorious (for now that is) with the advent of public key cryptography. This paper surveys the mathematical foundations, shortcomings, and novel variants of the “first” public key cryptosystem envisioned by Whitfield Diffie, Martin Hellman, and Ralph Merkle in 1976. The system they developed, Diffie-Hellman key exchange, relied on the difficulty of taking discrete logarithms in the finite fields Zp, where p is prime. While relatively secure, methods known as the index calculus exist to crack Diffie-Hellman key exchange in less than exponential running time. This has led to the use of elliptic curves in analogous cryptosystems. The basic theory underlying these elliptic curve cryptosystems is presented as well as a comparison of these systems with standard RSA encryption.