In this research, the choice of the parameter for a method to generate distinguished rational points in Pollard's Rho method to solve the elliptic curve discrete logarithm problem for Barreto-Naehrig (BN) curves is shown. The structures of random walk paths are confirmed by experiments for several BN curves. From the results, the authors clarify the conditions in which the Rho method does not stop during an attack, and the authors also show an indication for the choice of the parameter for the method to generate distinguished points with large bits of ECDLP.