Discrete Logarithms on Elliptic Curves

Cryptographic protocols often make use of the inherent hardness of the classical discrete logarithm problem, which is to solve gx = y (mod p) for x. The hardness of this problem has been exploited in the Diffie-Hellman key exchange, as well as in cryptosystems such as ElGamal. There is a similar discrete logarithm problem on elliptic curves: solve kB = P for k. Therefore, Diffie-Hellman and ElGamal have been adapted for elliptic curves. There is an abundance of evidence suggesting that elliptic curve cryptography is even more secure, which means that we can obtain the same security with fewer bits. In this paper, we investigate the discrete logarithm for elliptic curves over Fp for p>3 by constructing a function and considering the induced functional graph and the implications for cryptography.