The use of the computer, and specifically the mathematics software package Maple, has played a central role in the authors' abstract algebra course because it provides their students with a way to see realistic examples of the topics they discuss without having to struggle with extensive computations. However, Maple does not provide the computer science students in the course with any additional practice writing code in the industrial programming languages that are emphasized so heavily in their major. Recently, Diana Alexander, a student in the course majoring in both mathematics and computer science, implemented one of the systems they had discussed using Java instead of Maple. The system Alexander elected to work on was the ElGamal public-key cryptosystem constructed using an elliptic curve group. This system is one of the significant current mathematical interest, as elliptic curves have recently figured prominently in several types of mathematical problems. In this article, the authors briefly describe elliptic curves, the operation that gives them the structure of an abelian group, and the ElGamal cryptosystem in which they can naturally be incorporated. They then briefly summarize the implementation of the system using Java that Alexander was able to accomplish successfully.