计算工业强度原根的有效多项式时间算法
Efficient polynomial time algorithms computing industrial-strength primitive roots
摘要：
E.巴赫，遵循T。Itoh，已经展示了如何构建一个小的数字集合模一个素数p，使得这个集合中至少有一个元素是$\pF{p}$ \cite{Bach:1997:sppr,Itoh:2001:PPR}的生成元。E.巴赫还建议，他的作品集至少有一半应该是发电机。我们在这里表明，一个轻微的变化，这一套确实可以包含一个比率的原始根接近1的必要。因此，我们推导出几个算法计算原根正确的概率非常高，在多项式时间。特别是，我们提出了一个渐进的$O^{\sim}(\sqrt{\frac{1}ε}log^1.5(p) + \log^2(p))$算法提供原始根的$p$的正确性概率大于$1-ε$和几个$O(log^α(p))$，$α\leq 5.23$算法计算“工业强度”原始根的概率，例如大于“硬件故障”的概率。
英文摘要：
E. Bach, following an idea of T. Itoh, has shown how to build a small set of numbers modulo a prime p such that at least one element of this set is a generator of $\pF{p}$\cite{Bach:1997:sppr,Itoh:2001:PPR}. E. Bach suggests also that at least half of his set should be generators. We show here that a slight variant of this set can indeed be made to contain a ratio of primitive roots as close to 1 as necessary. We thus derive several algorithms computing primitive roots correct with very high probability in polynomial time. In particular we present an asymptotically $O^{\sim}(\sqrt{\frac{1}ε}log^1.5(p) + \log^2(p))$ algorithm providing primitive roots of $p$ with probability of correctness greater than $1-ε$ and several $O(log^α(p))$, $α\leq 5.23$ algorithms computing "Industrial-strength" primitive roots with probabilities e.g. greater than the probability of "hardware malfunctions".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
