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# Factoring with Cyclotomic Polynomials

Eric Bach and Jeffrey Shallit
Mathematics of Computation
Vol. 52, No. 185 (Jan., 1989), pp. 201-219
DOI: 10.2307/2008664
Stable URL: http://www.jstor.org/stable/2008664
Page Count: 19
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## Abstract

This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials $f(X)$ known to have the following property: given a multiple of $f(p)$, we can quickly split any composite number that has $p$ as a prime divisor. For example-taking $f(X)$ to be $X - 1$-a multiple of $p - 1$ will suffice to easily factor any multiple of $p$, using an algorithm of Pollard. Other methods (due to Guy, Williams, and Judd) make use of $X + 1, X^2 + 1$, and $X^2 \pm X + 1$. We show that one may take $f$ to be $\Phi_k$, the $k$th cyclotomic polynomial. In contrast to the ad hoc methods used previously, we give a universal construction based on algebraic number theory that subsumes all the above results. Assuming generalized Riemann hypotheses, the expected time to factor $N$ (given a multiple $E$ of $\Phi_k(p)$) is bounded by a polynomial in $k, \log E$, and $\log N$.

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