Pollard Rho碰撞的谱分析
Spectral Analysis of Pollard Rho Collisions
摘要：
我们证明了经典的Pollard rho算法在期望时间O（sqrt（n）（log n）^3）内产生碰撞。这是第一个非平凡的严格估计的碰撞概率不变的Pollard rho图，并接近于O（sqrt（n））的约束最优界。这个结果是通过证明这个图上的随机游动的混合时间是O（（log n）^3）而得出的;如果没有Pollard rho算法中的平方步骤，混合时间将是log n的指数。该技术涉及有向图的谱分析，其捕获平方步骤的效果。
英文摘要：
We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho graph, and is close to the conjectured optimal bound of O(sqrt(n)). The result is derived by showing that the mixing time for the random walk on this graph is O((log n)^3); without the squaring step in the Pollard rho algorithm, the mixing time would be exponential in log n. The technique involves a spectral analysis of directed graphs, which captures the effect of the squaring step.
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