ABSTRACT
In this chapter, the authors present a public-key cryptosystem whose security is based on the difficulty an intruder would encounter when solving the discrete logarithm problem. This system, called the ElGamal cryptosystem in honor of Taher Elgamal who first published the system in 1985, has formed an important area of recent cryptographic research due to how elliptic curves can naturally be incorporated into the system. Before describing elliptic curves and how they can naturally be incorporated into the ElGamal system, the authors also present the system in general and give a couple of simple examples of the system. Elliptic curves have figured prominently in several important mathematical problems. For example, the recent proof of Fermat’s Last Theorem by Andrew Wiles employed elliptic curves. The authors deal with one additional fact regarding elliptic curves. They show how MATLAB can be used to perform the computations.
