Cryptology ePrint Archive: Report 2012/294

Two grumpy giants and a baby

Daniel J. Bernstein and Tanja Lange

Abstract: Pollard's rho algorithm, along with parallelized, vectorized, and negating variants, is the standard method to compute discrete logarithms in generic prime-order groups. This paper presents two reasons that Pollard's rho algorithm is farther from optimality than generally believed. First, ``higher-degree local anti-collisions'' make the rho walk less random than the predictions made by the conventional Brent--Pollard heuristic. Second, even a truly random walk is suboptimal, because it suffers from ``global anti-collisions'' that can at least partially be avoided. For example, after (1.5+o(1))\sqrt(l) additions in a group of order l (without fast negation), the baby-step-giant-step method has probability 0.5625+o(1) of finding a uniform random discrete logarithm; a truly random walk would have probability 0.6753\ldots+o(1); and this paper's new two-grumpy-giants-and-a-baby method has probability 0.71875+o(1).

Category / Keywords: public-key cryptography / Discrete logarithms, Pollard rho, baby-step giant-step, anticollisions

Publication Info: Expanded version of ANTS 2012 paper.

Date: received 2 Jun 2012, last revised 10 Jul 2012

Contact author: tanja at hyperelliptic org

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Version: 20120710:183329 (All versions of this report)

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