Factoring N = prqs for Large r and s

Boneh et al. showed at Crypto 99 that moduli of the form N = p(r)q can be factored in polynomial time when r similar or equal to 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 with k prime factors N = Pi(k)(i=1) p(i)(ri); 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.