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.