# Factoring \(N=p^rq^s\) for Large *r* and *s*

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## Abstract

Boneh *et al.* showed at Crypto 99 that moduli of the form \(N=p^rq\) can be factored in polynomial time when \(r \simeq \log p\). Their algorithm is based on Coppersmith’s technique for finding small roots of polynomial equations. In this paper we show that \(N=p^rq^s\) can also be factored in polynomial time when *r* or *s* is at least \((\log p)^3\); therefore we identify a new class of integers that can be efficiently factored.

We also generalize our algorithm to moduli with *k* prime factors \(N= \prod _{i=1}^{k} p_i^{r_i}\); we show that a non-trivial factor of *N* can be extracted in polynomial-time if one of the exponents \(r_i\) is large enough.

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