. In Pollard's rho method, an iterating function f is used to define a sequence (y i ) by y i+1 = f(y i ) for i = 0; 1; 2; : : : , with some starting value y 0 . In this paper, we define 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 defined 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. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. Given an element h in G, we wish to find the least non-negative number x such that g x = h. This problem is the discre...

. We consider Pollard's rho method for discrete logarithm computation. Usually, in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated. We show that this assumption does not hold for the walk originally suggested by Pollard: its performance is worse than in the random case. We study alternative walks that can be efficiently applied to compute discrete logarithms. We introduce a class of walks that lead to the same performance as expected in the random case. We show that this holds for arbitrarily large prime group orders, thus making Pollard's rho method for prime group orders about 20% faster than before. 1. Introduction Let G be a finite cyclic group, written multiplicatively, and generated by the group element g. We define the discrete logarithm problem (DLP) as follows: given a group element h, find the least non-negative integer x such that h = g x . We write x = log g h and call it the discrete logarithm of h...

this paper, we give an algorithm which uses the Pohlig-Hellman method to find such a solution (y; x). Our algorithm has the advantage that apart from an O(log jGj) term, its run time is the 0747--7171/90/000000 + 00 $03.00/0 c fl 1999 Academic Press Limited 2 EDLYN TESKE