Elliptic curves have been studied for more than one hundred years. Their practical aspects in public key cryptography were discovered in mid-1980's. Since then elliptic curve cryptography has gained wide acceptance and been standardized. Efficiency, throughput or speed of the cryptosystem is a critical measure in commercial applications. Double-base number system approach has been used in improving the speed of elliptic curve cryptosystems. This book covers following topics: basics of public key cryptography, related field arithmetic, basics of elliptic curve cryptography, double-base number system and software/hardware implementation strategies for speeding up elliptic curve cryptosystems.