离散对数Pollard Rho碰撞的近最优界
Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
摘要：
本文分析了求循环群G中离散对数的Pollard Rho算法的一个相当标准的理想化。发现碰撞以$O(\sqrt{|G|\log |G| \log \log |G|})$步长发生的概率很高，与广泛确定的$Θ(\sqrt{|G|})$值相差不远。这改进了Miller-Venkatesan最近的一个结果，该结果给出了$O(\sqrt{|G|}\log^3 |G|)$的上界。我们的证明是基于分析一个适当的不可逆的，非懒惰的随机游动（奇数）长度|G|的离散周期，并显示相应的行走的混合时间是$O(\log |G| \log \log |G|)$。
英文摘要：
We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in $O(\sqrt{|G|\log |G| \log \log |G|})$ steps, not far from the widely conjectured value of $Θ(\sqrt{|G|})$. This improves upon a recent result of Miller--Venkatesan which showed an upper bound of $O(\sqrt{|G|}\log^3 |G|)$. Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is $O(\log |G| \log \log |G|)$.
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