The research on the attack algorithm for a MD5 collision is one of the focuses in cryptology nowadays. In this paper, by analyzing the properties of the nonlinear Boolean functions used in MD5 and the differences in term of XOR and subtraction modulo 232, we prove that some sufficient conditions presented by Jie Liang and Xuejia Lai are also necessary to guarantee the differential path and give a set of necessary and sufficient conditions to guarantee the output differences of the last two steps. Then we present an improved collision attack algorithm on MD5 by using the set of necessary and sufficient conditions. Finally, we analyze the computational complexity of our attack algorithm which is 0.718692 times of that of the previous collision attack algorithms.