We formally show that 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.
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