A deterministic version of Pollard’s $p-1$ algorithm
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Abstract:
In this article we present applications of smooth numbers to the unconditional derandomization of some well-known integer factoring algo- rithms.
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 $\varphi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\varphi (n)$. 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 $\varphi$ are sufficient to completely factor any integer $n$ in less than $\exp \Bigl ((1+o(1))(\ln n)^{\frac {1}{3}} (\ln \ln n)^{\frac {2}{3}}\Bigr )$ deterministic time.
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Additional Information
- Bartosz Źrałek
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, 00-956 Warsaw, Poland
- Email: b.zralek@impan.gov.pl
- Received by editor(s): November 26, 2007
- Received by editor(s) in revised form: October 3, 2008, and January 1, 2009
- Published electronically: May 6, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Math. Comp. 79 (2010), 513-533
- MSC (2000): Primary 11Y16; Secondary 11Y05, 68Q10
- DOI: https://doi.org/10.1090/S0025-5718-09-02262-5
- MathSciNet review: 2552238