Comparison Study on Point Counting Algorithms of Elliptic Curves Over Prime Field
Over hundred years ago, the arithmetic of elliptic curve had been studied for its theoretical reasons and not ever realized that elliptic curve had an essential application in cryptosystem until Victor Miller and Neal Koblitz independently proposed the public key cryptosystem which was based on elliptic curve. Elliptic curve can be defined over real field, complex field, rational field and finite field. However, the elliptic curve defined over finite field with characteristic not equal to two and three is the main interest and study object for mathematicians and cryptographers. Elliptic curve cryptosystem is freed from the sub-exponential attack and therefore the discrete logarithm of the group of elliptic curve is believed to be hard and infeasible.