Selecting a suitable model of an elliptic curve to have an elliptic curve cryptosystem that computes scaler multiplication as fast as possible, depends on different factors. In other words, it is completely necessary to know which model should be selected because it directly influences on security and performance of the elliptic curves cryptosystem. In this thesis, different models of elliptic curves have been discussed. Moreover, I have considered and analyzed different aspects to find models of elliptic curve that satisfy both security and efficiency. My suggestion based on the experiment is the prime order elliptic curves in Weierstrass model, Montgomery model and twisted Edwards model that have their practical advantages. In addition, these curves are suitable for current implementations that are supported by National Institute of Standards and Technology (NIST) over prime finite fields. It could be confederated into existing implementations by changing the field arithmetic and the curve constant in some cases. My thesis also have examined different prime finite fields to find the most efficient modular arithmetic. For example, for modular arithmetic operations, using Montgomery friendly primes and Pseudo Mersene primes enhance the performance of the elliptic curve cryptosystem. In addition, to have a faster computations we require to select primes that are congruent to ? modulo ?.
