Speeding Up Pollard's Rho Method for Computing Discrete Logarithms

In Pollard's rho method, an iterating function f is used to de-ne a sequence (yi) by yi+1 = f(yi) for i = 0; 1; 2; : : :, with some starting value y0. In this paper, we deene and discuss new iterating functions for computing discrete logarithms with the rho method. We compare their performances in experiments with elliptic curve groups. Our experiments show that one of our newly deened functions is expected to reduce the number of steps by a factor of approximately 0:8, in comparison with Pollard's originally used function, and we show that this holds independently of the size of the group order. For group orders large enough such that the run time for precomputation can be neglected, this means a real-time speed-up of more than 1:2.