**Factoring N=p^r q^s for Large r and s**

*Jean-Sebastien Coron and Jean-Charles Faugere and Guenael Renault and Rina Zeitoun*

**Abstract: **Boneh et al. showed at Crypto 99 that moduli of the form N=p^r q can be factored in polynomial time when r=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^r q^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 N with k prime factors; we show that a non-trivial factor of N can be extracted in polynomial-time if one of the k exponents is large enough.

**Category / Keywords: **public-key cryptography / Factoring N=p^r q^s, Coppersmith's Algorithm, LLL, RSA.

**Original Publication**** (with major differences): **CT-RSA 2016

**Date: **received 1 Feb 2015, last revised 24 Nov 2015

**Contact author: **jean-sebastien coron at uni lu

**Available format(s): **PDF | BibTeX Citation

**Version: **20151124:102608 (All versions of this report)

**Short URL: **ia.cr/2015/071

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