Abstract
We investigate the problem of factoring RSA moduli with implicit hint, which was firstly proposed by May and Ritzenhofen in 2009 where unknown prime factors of several RSA moduli shared some number of least significant bits (LSBs) and was considered by Faug\(\grave{e}\)re et al. in 2010 where some most significant bits (MSBs) were shared between the primes. In this paper, we further consider this factorization with implicit hint problem, present a method to deal with the case when the number of shared LSBs or MSBs is not large enough to satisfy the bound proposed by May et al. and Faug\(\grave{e}\)re et al. by making use of a result from Herrmann and May for solving linear equations modulo unknown divisors, and finally get a better lower bound on the the number of shared LSBs or MSBs. To the best of our knowledge, our lower bound is better than all known results and we can theoretically deal with the implicit factorization for the case of balanced RSA moduli.
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References
Ajtai, M.: Generating random lattices according to the invariant distribution. Draft of March (2006)
Ajtai, M.: The shortest vector problem in L 2 is NP-hard for randomized reductions. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 10–19. ACM (1998)
Bernstein, D.J., Chang, Y.-A., Cheng, C.-M., Chou, L.-P., Heninger, N., Lange, T., van Someren, N.: Factoring RSA keys from certified smart cards: Coppersmith in the wild. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 341–360. Springer, Heidelberg (2013)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N 0.292. IEEE Transactions on Information Theory 46(4), 1339–1349 (2000)
Cohn, H., Heninger, N.: Approximate common divisors via lattices. arXiv preprint arXiv:1108.2714 (2011)
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. Journal of Cryptology 10(4), 233–260 (1997)
Faugère, J.-C., Marinier, R., Renault, G.: Implicit factoring with shared most significant and middle bits. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 70–87. Springer, Heidelberg (2010)
Herrmann, M., May, A.: Solving linear equations modulo divisors: On factoring given any bits. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 406–424. Springer, Heidelberg (2008)
Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)
Howgrave-Graham, N.: Approximate integer common divisors. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 51–66. Springer, Heidelberg (2001)
Jochemsz, E., May, A.: A strategy for finding roots of multivariate polynomials with new applications in attacking RSA variants. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 267–282. Springer, Heidelberg (2006)
Jochemsz, E., May, A.: A polynomial time attack on RSA with private CRT-exponents smaller than N 0.073. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 395–411. Springer, Heidelberg (2007)
Lenstra, A.K., Hughes, J.P., Augier, M., Bos, J.W., Kleinjung, T., Wachter, C.: Public keys. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 626–642. Springer, Heidelberg (2012)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534 (1982)
Lu, Y., Zhang, R., Lin, D.: Improved bounds for the implicit factorization problem. Advances in Mathematics of Communications 7(3), 243–251 (2013)
May, A., Ritzenhofen, M.: Implicit factoring: On polynomial time factoring given only an implicit hint. In: Jarecki, S., Tsudik, G. (eds.) PKC 2009. LNCS, vol. 5443, pp. 1–14. Springer, Heidelberg (2009)
Nguên, P.Q., Stehlé, D.: Floating-point LLL revisited. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005)
Nguyen, P.Q., Valle, B.: The LLL algorithm: Survey and applications. Springer Publishing Company, Incorporated (2009)
Sarkar, S., Maitra, S.: Approximate integer common divisor problem relates to implicit factorization. IEEE Transactions on Information Theory 57(6), 4002–4013 (2011)
Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36(3), 553–558 (1990)
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Peng, L., Hu, L., Xu, J., Huang, Z., Xie, Y. (2014). Further Improvement of Factoring RSA Moduli with Implicit Hint. In: Pointcheval, D., Vergnaud, D. (eds) Progress in Cryptology – AFRICACRYPT 2014. AFRICACRYPT 2014. Lecture Notes in Computer Science, vol 8469. Springer, Cham. https://doi.org/10.1007/978-3-319-06734-6_11
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DOI: https://doi.org/10.1007/978-3-319-06734-6_11
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