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

Jean-Sébastien Coron (1) , Jean-Charles Faugère (2) , Guénaël Renault (2) , Rina Zeitoun (3, 2)

1 Uni.lu - Université du Luxembourg
2 PolSys - Polynomial Systems
3 Oberthur Card Systems - Puteaux

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 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.

Domaines: Calcul formel [cs.SC]; Cryptographie et sécurité [cs.CR]

Jean-Sébastien Coron, Jean-Charles Faugère, Guénaël Renault, Rina Zeitoun. Factoring $N=p^r q^s$ for Large $r$ and $s$. RSA Conference Cryptographers' Track , Feb 2016, San Francisco, United States. ⟨10.1007/978-3-319-29485-8_26⟩. ⟨hal-01250302⟩