February 2019, 13(1): 121-135. doi: 10.3934/amc.2019007

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

1. 

College of Computer, National University of Defense Technology, Changsha 410073, China

2. 

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore

3. 

State Key Laboratory of Cryptology, Beijing 100878, China

4. 

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

* Corresponding author: Longjiang Qu

Received  May 2018 Revised  September 2018 Published  December 2018

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 $N = p_1^{r_1}p_2^{r_2}··· p_n^{r_n}$.

Citation: Shixiong Wang, Longjiang Qu, Chao Li, Huaxiong Wang. Further improvement of factoring $ N = p^r q^s$ with partial known bits. Advances in Mathematics of Communications, 2019, 13 (1) : 121-135. doi: 10.3934/amc.2019007
References:
[1]

D. Boneh, G. Durfee and N. Howgrave-Graham, Factoring $ N = p^r q$ for large $ r$, Advances in Cryptology-CRYPTO 1999, Springer Berlin Heidelberg, 1666 (1999), 326-337. doi: 10.1007/3-540-48405-1_21.

[2]

T. Collins, D. Hopkins, S. Langford and M. Sabin, Public key cryptographic apparatus and method, U.S. Patent, (1998), 5848159.

[3]

D. Coppersmith, Finding a small root of a univariate modular equation, Advances in Cryptology-EUROCRYPT 1996, Springer Berlin Heidelberg, 1070 (1996), 155-165. doi: 10.1007/3-540-68339-9_14.

[4]

D. Coppersmith, Finding a small root of a bivariate integer equation; factoring with high bits known, Advances in Cryptology-EUROCRYPT 1996, Springer Berlin Heidelberg, 1070 (1996), 178-189. doi: 10.1007/3-540-68339-9_16.

[5]

D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities, Journal of Cryptology, 10 (1997), 233-260. doi: 10.1007/s001459900030.

[6]

J. S. Coron, J. C. Faugère, G. Renault and R. Zeitoun, Factoring $ N = p^rq^s$ for large $ r$ and $ s$, Cryptographers' Track at the RSA Conference, Springer, Cham, 9610 (2016), 448-464. doi: 10.1007/978-3-319-29485-8_26.

[7]

J. S. Coron and R. Zeitoun, Improved factorization of $ N = p^rq^s$, Cryptographers' Track at the RSA Conference, Springer, Cham, 2018, 65-79.

[8]

M. Herrmann and A. May, On factoring arbitrary integers with known bits, IACR Cryptology ePrint Archive, 2007, 374, https://eprint.iacr.org/2007/374.

[9]

M. J. Hinek, On the security of multi-prime RSA, Journal of Mathematical Cryptology, 2 (2008), 117-147. doi: 10.1515/JMC.2008.006.

[10]

N. Howgrave-Graham, Finding small roots of univariate modular equations revisited, Crytography and Coding, Springer Berlin Heidelberg, 1335 (1997), 131-142. doi: 10.1007/BFb0024458.

[11]

N. Howgrave-Graham, Approximate integer common divisors, Cryptography and Lattices, Springer Berlin Heidelberg, 2146 (2001), 51-66. doi: 10.1007/3-540-44670-2_6.

[12]

A. K. LenstraH. W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen, 261 (1982), 515-534. doi: 10.1007/BF01457454.

[13]

S. Lim, S. Kim, I. Yie and H. Lee, A generalized Takagi-cryptosystem with a modulus of the form $ p^r q^s$, Progress in Cryptology-INDOCRYPT 2000, Springer Berlin Heidelberg, 1977 (2000), 283-294. doi: 10.1007/3-540-44495-5_25.

[14]

Y. Lu, L. Peng and S. Sarkar, Cryptanalysis of an RSA variant with moduli $ N = p^rq^l$, The 9th International Workshop on Coding and Cryptography 2015, WCC 2015.

[15]

A. May, New RSA Vulnerabilities Using Lattice Reduction Methods, Ph.D. thesis, University of Paderborn, 2003.

[16]

R. L. RivestA. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, 21 (1978), 120-126. doi: 10.1145/359340.359342.

[17]

B. Santoso, N. Kunihiro, N. Kanayama and K. Ohta, Factorization of square-free integers with high bits known, Progress in Cryptology-VIETCRYPT 2006, Springer, Berlin, Heidelberg, 2006,115-130.

[18]

T. Takagi, Fast RSA-type cryptosystem modulo $ p^k q$, Advances in Cryptology-CRYPTO 1998, Springer Berlin Heidelberg, 1998,318-326.

[19]

H. Zhang and T. Takagi, Attacks on multi-prime RSA with small prime difference, Australasian Conference on Information Security and Privacy, Springer, Berlin, Heidelberg, 2013, 41-56.

[20]

M. Zheng, N. Kunihiro and H. Hu, Improved factoring attacks on multi-prime RSA with small prime difference, Australasian Conference on Information Security and Privacy, Springer, Cham, 2017,324-342.

[21]

SageMath, the Sage Mathematics Software System, the Sage Developers, 2018, http://www.sagemath.org.

show all references

References:
[1]

D. Boneh, G. Durfee and N. Howgrave-Graham, Factoring $ N = p^r q$ for large $ r$, Advances in Cryptology-CRYPTO 1999, Springer Berlin Heidelberg, 1666 (1999), 326-337. doi: 10.1007/3-540-48405-1_21.

[2]

T. Collins, D. Hopkins, S. Langford and M. Sabin, Public key cryptographic apparatus and method, U.S. Patent, (1998), 5848159.

[3]

D. Coppersmith, Finding a small root of a univariate modular equation, Advances in Cryptology-EUROCRYPT 1996, Springer Berlin Heidelberg, 1070 (1996), 155-165. doi: 10.1007/3-540-68339-9_14.

[4]

D. Coppersmith, Finding a small root of a bivariate integer equation; factoring with high bits known, Advances in Cryptology-EUROCRYPT 1996, Springer Berlin Heidelberg, 1070 (1996), 178-189. doi: 10.1007/3-540-68339-9_16.

[5]

D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities, Journal of Cryptology, 10 (1997), 233-260. doi: 10.1007/s001459900030.

[6]

J. S. Coron, J. C. Faugère, G. Renault and R. Zeitoun, Factoring $ N = p^rq^s$ for large $ r$ and $ s$, Cryptographers' Track at the RSA Conference, Springer, Cham, 9610 (2016), 448-464. doi: 10.1007/978-3-319-29485-8_26.

[7]

J. S. Coron and R. Zeitoun, Improved factorization of $ N = p^rq^s$, Cryptographers' Track at the RSA Conference, Springer, Cham, 2018, 65-79.

[8]

M. Herrmann and A. May, On factoring arbitrary integers with known bits, IACR Cryptology ePrint Archive, 2007, 374, https://eprint.iacr.org/2007/374.

[9]

M. J. Hinek, On the security of multi-prime RSA, Journal of Mathematical Cryptology, 2 (2008), 117-147. doi: 10.1515/JMC.2008.006.

[10]

N. Howgrave-Graham, Finding small roots of univariate modular equations revisited, Crytography and Coding, Springer Berlin Heidelberg, 1335 (1997), 131-142. doi: 10.1007/BFb0024458.

[11]

N. Howgrave-Graham, Approximate integer common divisors, Cryptography and Lattices, Springer Berlin Heidelberg, 2146 (2001), 51-66. doi: 10.1007/3-540-44670-2_6.

[12]

A. K. LenstraH. W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Mathematische Annalen, 261 (1982), 515-534. doi: 10.1007/BF01457454.

[13]

S. Lim, S. Kim, I. Yie and H. Lee, A generalized Takagi-cryptosystem with a modulus of the form $ p^r q^s$, Progress in Cryptology-INDOCRYPT 2000, Springer Berlin Heidelberg, 1977 (2000), 283-294. doi: 10.1007/3-540-44495-5_25.

[14]

Y. Lu, L. Peng and S. Sarkar, Cryptanalysis of an RSA variant with moduli $ N = p^rq^l$, The 9th International Workshop on Coding and Cryptography 2015, WCC 2015.

[15]

A. May, New RSA Vulnerabilities Using Lattice Reduction Methods, Ph.D. thesis, University of Paderborn, 2003.

[16]

R. L. RivestA. Shamir and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, 21 (1978), 120-126. doi: 10.1145/359340.359342.

[17]

B. Santoso, N. Kunihiro, N. Kanayama and K. Ohta, Factorization of square-free integers with high bits known, Progress in Cryptology-VIETCRYPT 2006, Springer, Berlin, Heidelberg, 2006,115-130.

[18]

T. Takagi, Fast RSA-type cryptosystem modulo $ p^k q$, Advances in Cryptology-CRYPTO 1998, Springer Berlin Heidelberg, 1998,318-326.

[19]

H. Zhang and T. Takagi, Attacks on multi-prime RSA with small prime difference, Australasian Conference on Information Security and Privacy, Springer, Berlin, Heidelberg, 2013, 41-56.

[20]

M. Zheng, N. Kunihiro and H. Hu, Improved factoring attacks on multi-prime RSA with small prime difference, Australasian Conference on Information Security and Privacy, Springer, Cham, 2017,324-342.

[21]

SageMath, the Sage Mathematics Software System, the Sage Developers, 2018, http://www.sagemath.org.

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 $
$\log_2 N$ $ 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
$\log_2 N$
$(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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