Toward a theory of Pollard's rho method
Information and Computation
On random walks for Pollard's Rho method
Mathematics of Computation
Speeding Up Pollard's Rho Method for Computing Discrete Logarithms
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
On the second eigenvalue of random regular graphs
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Randomness-efficient oblivious sampling
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Lower bounds for discrete logarithms and related problems
EUROCRYPT'97 Proceedings of the 16th annual international conference on Theory and application of cryptographic techniques
On the efficiency of Pollard's rho method for discrete logarithms
CATS '08 Proceedings of the fourteenth symposium on Computing: the Australasian theory - Volume 77
Spectral analysis of pollard rho collisions
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
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We formally showthat there is an algorithm for dlog over all abelian groups that runs in expected optimal time (up to logarithmic factors) and uses only a small amount of space. To our knowledge, this is the first such analysis. Our algorithm is a modification of the classic Pollard rho, introducing explicit randomization of the parameters for the updating steps of the algorithm, and is analyzed using random walks with limited independence over abelian groups (a study which is of its own interest). Our analysis shows that finding cycles in such large graphs over groups that can be efficiently locally navigated is as hard as DLOG.