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A better bound for implicit factorization problem with shared middle bits

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Abstract

This paper presents our investigation of the implicit factorization problem, where unknown prime factors of two RSA moduli share a certain number of middle bits. The problem is described as follows. Let N 1 = p 1 q 1, N 2 = p 2 q 2 be two different n-bit RSA moduli, where q 1, q 2 are both αn-bit prime integers. Suppose that p 1, p 2 share tn bits at positions from t 1 n to t 2 n = (t 1 + t)n. Then this problem focuses on the condition about t, α to factor N 1,N 2 efficiently. At PKC 2010, Faugère et al. showed that N 1,N 2 can be factored when t > 4α. Subsequently, in 2015, Peng et al. improved this bound to t > 4α−3α 2. In this paper, we directly apply Coppersmith’s method to the implicit factorization problem with shared middle bits, and a better bound \(t > 4\alpha - 4{\alpha ^{\frac{3}{2}}}\) is obtained. The correctness of our approach is verified by experiments.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11531002, 61572026), Basic Research Fund of National University of Defense Technology (Grant No. CJ 13-02-01), Open Foundation of State Key Laboratory of Cryptology, and Program for New Century Excellent Talents in University (NCET).

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Authors and Affiliations

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

    Shixiong Wang, Chao Li & Shaojing Fu

  2. State Key Laboratory of Cryptology, Beijing, 100878, China

    Longjiang Qu & Shaojing Fu

  3. College of Science, National University of Defense Technology, Changsha, 410073, China

    Longjiang Qu & Chao Li

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  1. Shixiong Wang
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  2. Longjiang Qu
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  3. Chao Li
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  4. Shaojing Fu
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Correspondence to Longjiang Qu.

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Wang, S., Qu, L., Li, C. et al. A better bound for implicit factorization problem with shared middle bits. Sci. China Inf. Sci. 61, 032109 (2018). https://doi.org/10.1007/s11432-017-9176-5

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  • DOI: https://doi.org/10.1007/s11432-017-9176-5

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