# 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

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

Preview not available

## 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 $\log N$.

• 201
• 202
• 203
• 204
• 205
• 206
• 207
• 208
• 209
• 210
• 211
• 212
• 213
• 214
• 215
• 216
• 217
• 218
• 219