离散对数Pollard Rho算法中具有最佳碰撞界的马尔可夫链的生日参数
A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm
摘要：
本文给出了均匀平稳分布的马尔可夫链的自相交的一个生日定理。作为一个应用，我们分析了Pollard的Rho算法在循环群$G$中求离散对数，发现如果算法中的划分是由随机预言机给出的，那么在$Θ(\sqrt{|G|})$步中很有可能发生碰撞.此外，对于$J$处理器上的并行化区别点算法，我们发现$Θ(\sqrt{|G|}/J)$步骤就足够了。这些是第一次证明的正确顺序界限，不假设每一步的算法产生的i.i.d.样品来自$G$。
英文摘要：
We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group $G$ and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in $Θ(\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on $J$ processors we find that $Θ(\sqrt{|G|}/J)$ steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from $G$.
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