Improved Factorization of \(N=p^rq^s\)
Boneh et al. showed at Crypto 99 that moduli of the form \(N=p^rq\) can be factored in polynomial time when \(r \ge \log p\). Their algorithm is based on Coppersmith’s technique for finding small roots of polynomial equations. Recently, Coron et al. showed that \(N=p^rq^s\) can also be factored in polynomial time, but under the stronger condition \(r \ge \log ^3 p\). In this paper, we show that \(N=p^rq^s\) can actually be factored in polynomial time when \(r \ge \log p\), the same condition as for \(N=p^rq\).
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