This master thesis is about Elliptic Curve Digital Signature Algorithm or ECDSA and two of the known attacks on this security system. The purpose of this thesis is to find points that are likely to be points of high order on an elliptic curve. If we have a point P of high order and if Q = mP, then we have a large set of possible values of m. Therefore it is hard to solve the Elliptic Curve Discrete Logarithm Problem or ECDLP. We have investigated on the time of finding the solution of ECDLP for a certain amount of elliptic curves based on the order of the point which is used to create the digital signatures by those elliptic curves. Method: Algebraic Structure of elliptic curves over finite fields and Discrete logarithms. This has been done by two types of attacks namely Baby Step, Giant Step and Pollard’s Rho and all of the programming parts has been done by means of Mathematica. Conclusion: We have come into a conclusion of having the probable good points which are the points of high order on elliptic curves through the mentioned attacks in which solving the ECDLP is harder if these points have been used in generating the digital signature. These probable good points can be estimated by means of a function we have come up with. The input of this function is the order of the point and the output is the time of finding the answer of ECDLP.