Abstract
The problem of factoring RSA moduli with the implicit hint was firstly proposed by May and Ritzenhofen at PKC’09 where unknown prime factors of several RSA moduli shared some number of least significant bits (LSBs), and was later considered by Faugère et al. where some most significant bits (MSBs) were shared between the primes. Recently, Nitaj and Ariffin proposed a generalization of the implicit factorization problem. Let \( {\text{N}}_{1} = {\text{p}}_{1} {\rm{q}}_{1} \) and \( {\text{N}}_{2} = {\text{p}}_{2} {\rm{q}}_{2} \) be two distinct RSA moduli, Nitaj and Ariffin showed that when \( {\text{a}}_{1} {\rm{p}}_{1} \) and \( {\text{a}}_{2} {\rm{p}}_{2} \) share enough bits, \( {\text{N}}_{1} , {\rm{N}}_{2} \) can be factored in polynomial time, where \( {\text{a}}_{1} \) and \( {\text{a}}_{2} \) are some unknown positive integers. They also extended their work to the case of \( k\left( { \ge 3} \right) \) moduli. In this paper, we revisit Nitaj-Ariffin’s work and transform the problem into solving small roots of a modular equation. Then by utilizing Coppersmith’s method, for the case of two moduli we improve Nitaj-Ariffin’s result when the unknowns \( {\text{a}}_{1} ,{\rm{a}}_{2} \) are relatively small, and our result is always better than Nitaj-Ariffin’s result for the case of \( k\left( { \ge 3} \right) \) moduli.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N0.292. IEEE Trans. Inf. Theory 46(4), 1339–1349 (2000)
Coppersmith, D.: Finding a small root of a univariate modular equation. In: EUROCRYPT 1996, pp. 155–165 (1996)
Coppersmith, D.: Finding a small root of a bivariate integer equation factoring with high bits known. In: EUROCRYPT 1996, pp. 178–189 (1996)
Faugère, J.-C., Mariner, R., Renault, G.: Implicit factoring with shared most significant and middle bits. In: PKC 2010, pp. 70–87 (2010)
Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Cryptography and Coding 1997, pp. 131–142 (1997)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)
Lu, Y., Peng, L., Zhang, R., Hu, L., Lin, D.: Towards optimal bounds for implicit factorization problem. In: SAC 2015, pp. 462–476 (2015)
Lu, Y., Zhang, R., Peng, L., Lin, D.: Solving linear equations modulo unknown divisors: Revisited. In: ASIACRYPT 2015, Part I, pp. 189–213 (2015)
May, A.: New RSA vulnerabilities using lattice reduction methods. Ph.D. thesis, University of Paderborn (2003). http://ubdata.uni-paderborn.de/ediss/17/2003/may/disserta.pdf
May, A., Ritzenhofen, M.: Implicit factoring: on polynomial time factoring given only an implicit hint. In: PKC 2009, pp. 1–14 (2009)
Nitaj, A., Ariffin, M.: Implicit factorization of unbalanced RSA moduli. J. Appl. Math. Comput. 48(1–2), 349–363 (2015)
Peng, L., Hu, L., Xu, J., Huang, Z., Xie, Y.: Further improvement of factoring RSA moduli with implicit hint. In: AFRICACRYPT 2014, pp. 165–177 (2014)
Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)
Sarkar, S., Maitra, S.: Approximate integer common divisor problem relates to implicit factorization. IEEE Trans. Inf. Theory 57(6), 4002–4013 (2011)
Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Trans. Inf. Theory 36(3), 553–558 (1990)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Sun, Z., Zhang, T., Zheng, X., Yang, L., Peng, L. (2019). A Method for Solving Generalized Implicit Factorization Problem. In: Sun, S., Fu, M., Xu, L. (eds) Signal and Information Processing, Networking and Computers. ICSINC 2018. Lecture Notes in Electrical Engineering, vol 550. Springer, Singapore. https://doi.org/10.1007/978-981-13-7123-3_34
Download citation
DOI: https://doi.org/10.1007/978-981-13-7123-3_34
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-7122-6
Online ISBN: 978-981-13-7123-3
eBook Packages: EngineeringEngineering (R0)