On random walks for Pollard’s rho method

Abstract

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="20"> <mml:semantics> <mml:mn>20</mml:mn> <mml:annotation encoding="application/x-tex">20%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> faster than before.