Elliptic Curve Cryptography

Abstract

The first proposed asymmetric encryption scheme was that of Rivest, Shamir, and Adleman, using exponentiation in the group of integers modulo the product of two large primes. Koblitz and Miller independently proposed the use of the groups of points on elliptic curves. In this chapter we cover the algorithm for using curves for cryptography both for encryption and for key exchange. Since the arithmetic to do point addition is expensive, we include the formulas for adding points efficiently. Finally, we include the Pohlig-Hellman attack, which should not be successful if the curves are chosen properly, and the Pollard rho attack, which is the current best attack on the elliptic curve discrete log problem.