Factoring with cyclotomic polynomials

Abstract

This paper discusses some new integer factoring methods involving cyclotomic polynomials. There are several polynomials f ( X ) f(X) known to have the following property: given a multiple of f ( p ) f(p) , we can quickly split any composite number that has p as a prime divisor. For example—taking f ( X ) f(X) to be X − 1 X - 1 —a multiple of p − 1 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 + 1 , X 2 + 1 {X^2} + 1 , and X 2 ± X + 1 {X^2} \pm X + 1 . We show that one may take f to be Φ k {\Phi _k} , the kth 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 Φ k ( p ) {\Phi _k}(p) ) is bounded by a polynomial in k, log ⁡ E \log E , and log ⁡ N \log N .