波拉德p-1算法的确定性版本
A deterministic version of Pollard's p-1 algorithm
摘要：
在这篇文章中，我们提出了光滑数的应用，一些著名的整数分解算法的无条件去随机化。我们从Pollard的$p-1$算法开始，该算法在随机多项式时间内找到整数$n$的素除子因子$p$，使得$p-1$是光滑的。我们表明，这些素因子可以恢复在确定性的多项式时间。我们进一步推广了这个结果，给出了由Bach和Shallit设计的$k$次分圆分解方法（$k\ge 2$）的部分去随机化。我们还研究了计算欧拉函数$ϕ$的因式分解的简化。我们指出了在给定$ϕ(n)$的确定性多项式时间内完全可因式分解的整数$n$的一些显式集合。这些集包括，粗略地说，产品的素数$p$满足，除了最多两个，某些条件稍弱于光滑的$p-1$。最后，我们证明了$O(\ln n)$ Oracle查询的$ϕ$值是足够的，以完全的因素，任何整数$n$小于$\exp\Bigl((1+o(1))(\ln n)^{1/3} (\ln\ln n)^{2/3}\Bigr)$确定性时间。
英文摘要：
In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algorithms. We begin with Pollard's $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ is smooth. We show that these prime factors can be recovered in deterministic polynomial time. We further generalize this result to give a partial derandomization of the $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $ϕ$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $ϕ(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $ϕ$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{1/3} (\ln\ln n)^{2/3}\Bigr)$ deterministic time.
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