The Elliptic curve discrete logarithm problem (ECDLP) is a significant problem in the epoch of modern cryptography. The security of various cryptographic systems and protocols, such as the Diffie-Hellman Key Exchange (DHKE), the US governments Digital Signature Algorithm (DSA) and, the Schnorr signature scheme, relies on the intractability of solving the Elliptic curve discrete logarithm problem. The elliptic curve cryptosystem (ECC) was proposed independently by Neal Koblitz and Victor Miller in 1985. In fact they utilize the group of points on an elliptic curve that is defined over finite prime field instead of the multiplicative groups of finite prime fields. Afterwards several practical systems based on the ECDLP were designed, such as various pairing based schemes. Nowadays, Pollard rho method and its parallelized versions are known as the best generic attack against the ECDLP. These methods can be used for any cyclic group and do not require to define any additional structure in the elliptic curve groups. This thesis explains a new iteration function for the rho method. This new method will be given by exploiting the fact that point halving is more efficient than point addition for elliptic curves over binary fields. We present a careful analysis of the rho method using the new iteration function, in comparison with the previous iteration functions. In generally the new method can achieve a significant speedup for computing ECDLP over binary fields. For instance, for certain NIST recommended curves over binary fields, the new method is about 12-17 % faster than the previous best methods.