Development of Solving the ECDLP

The study and analysis of the Elliptic curve Discrete Logarithm Problem (ECDLP) is very important, because one of the strongest cryptosystems is based on this problem, which is elliptic curve cryptosystem. This aspect has led to make a clear change in cryptography and open new research field for treatment with special group that is the group of elliptic curve points.This paper proposes a method for solving ECDLP, depending on an important theorem in abstract algebra. This method works in special groups, which are the cyclic group of elliptic curve points. As a result, we found a criterion that gives additional strength to the elliptic curve cryptosystems, which is the choice of the private key