Elliptic curve cryptography: Algorithms and implementation analysis over coordinate systems

Since the last decade, the growth of computing power and parallel computing has resulted in significant needs of efficient cryptosystem. Elliptic Curve Cryptography (ECC) offers faster computation and stronger security over other asymmetric cryptosystems such as RSA. ECC can be used for several cryptography activities: secret key sharing, message encryption, and digital signature. This paper gives step-by-step tutorial to transform ECC over prime field GF(p) from mathematical concept to the software implementation. This paper also gives several alternatives and tradeoffs between different coordinate systems in the computational process. The implementation result is quite interesting since several computational costs have been optimized in latest instruction sets. For the study case, we provides the implementation result in C language with GNU GMP library on Intel i3 CPU M350 2.27GHz (1 Core, 2 GB RAM, 32-bit architecture).