This book provides a comprehensive and self-contained introduction to elliptic curve and how they are employed to construct secure public key kryptosystems. Even though the elegant mathematical theory underlying cryptosystems is considerably more involved than for other systems, this text requires the reader to have only an elementary knowledge of basic algebra. The text nevertheless leads to problems at the forefront of the current research, featuring chapters on the point counting algorithms and security issues. The adopted unifying approach treats with equal care elliptic curves over fields of even characteristics, which are especially suited for hardware implementations, and curves over fields of odd characteristics, which have traditionally received more attention. This book will be of great interest to mathematicians, computer scientists, and engineers who are curious about elliptic curve cryptography in practise, without losing the beauty of the underlying mathematics.