Article Contents
Article Contents

# Further improvement of factoring $N = p^r q^s$ with partial known bits

• * Corresponding author: Longjiang Qu
• We revisit the factoring with known bits problem on RSA moduli. In 1996, Coppersmith showed that the RSA modulus $N = pq$ with balanced $p,q$ can be efficiently factored, if the high order $\frac{1}{4} \log_2 N$ bits of one prime factor is given. Later, this important result is also generalized to the factorization of RSA variants moduli such as $N = p^r q$ or $N = p_1 p_2 ··· p_n$. In 2000, Lim et al. proposed a new RSA variant with the modulus of the form $N = p^r q^s$, which is much faster in the decryption process than the standard RSA. Then from 2015 to 2018, in order to investigate the security property of this RSA variant, Lu et al. and Coron et al. have presented three works studying the polynomial-time factorization of $N = p^r q^s$ with partial known bits of $p^u q^v$ (or one of the prime factors $p,q$) for different choices of $u, v$. In this paper, we present a new lattice construction used for Coppersmith's method, and thus improve previous results. Namely, our result requires fewer known bits to recover the prime factors $p,q$. We also generalize our result to the factorization of $N = p_1^{r_1}p_2^{r_2}··· p_n^{r_n}$.

Mathematics Subject Classification: Primary: 11Y05; Secondary: 94A60.

 Citation:

• Table 1.  Comparison between the Result in [7] and Our Result when $\log_2 p \approx \log_2 q$

 $\log_2 N$ $r, s$ the Choice of $u, v$ the Result in [7] Our Result $5000$ $5, 4$ $4, 3$ $\zeta > 108.0$ $\zeta > 61.73$ $5000$ $7, 3$ $5, 2$ $\zeta > 70.00$ $\zeta > 50.00$ $2000$ $3, 2$ $2, 1$ $\zeta > 120.0$ $\zeta > 80.00$ $2000$ $7, 5$ $3, 2$ $\zeta > 23.15$ $\zeta > 13.89$ $10000$ $7, 5$ $3, 2$ $\zeta > 115.7$ $\zeta > 69.44$ $10000$ $13, 12$ $12, 11$ $\zeta > 30.67$ $\zeta > 16.00$

Table 2.  Experimental Examples for the Method in [7] and Our Method when $\log_2 p \approx \log_2 q$

 $\log_2 N$$(r,s),\ (u,v)$ Methods Bits Required At Least $m,t_1,t_2$ $\dim(\Lambda)$ Time (LLL Algorithm) $2000$ [7] $222$ bits $10,9,9$ $11$ $0.189$ seconds $(3,2),\ (2,1)$ Ours $196$ bits $10,9,10$ $11$ $0.170$ seconds $2000$ [7] $179$ bits $20,18,18$ $21$ $15.98$ seconds $(3,2),\ (2,1)$ Ours $141$ bits $20,18,20$ $21$ $8.517$ seconds $2000$ [7] $162$ bits $30,27,27$ $31$ $264.6$ seconds $(3,2),\ (2,1)$ Ours $118$ bits $30,27,30$ $31$ $94.78$ seconds $1000$ [7] $33$ bits $36,35,35$ $37$ $36.82$ seconds $(7,5),\ (3,2)$ Ours $30$ bits $36,35,36$ $37$ $31.85$ seconds $2000$ [7] $132$ bits $16,15,15$ $17$ $0.906$ seconds $(5,3),\ (2,1)$ Ours $120$ bits $16,15,16$ $17$ $0.557$ seconds $3000$ [7] $191$ bits $21,20,20$ $22$ $35.49$ seconds $(4,3),\ (3,2)$ Ours $161$ bits $21,20,21$ $22$ $21.91$ seconds
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