Abstract
Given two integers \(N_1 = p_1q_1\) and \(N_2 = p_2q_2\) with \(\alpha \)-bit primes \(q_1,q_2\), suppose that the \(t\) least significant bits of \(p_1\) and \(p_2\) are equal. May and Ritzenhofen (PKC 2009) developed a factoring algorithm for \(N_1,N_2\) when \(t \ge 2\alpha + 3\); Kurosawa and Ueda (IWSEC 2013) improved the bound to \(t \ge 2\alpha + 1\). In this paper, we propose a polynomial-time algorithm in a parameter \(\kappa \), with an improved bound \(t = 2\alpha - O(\log \kappa )\); it is the first non-constant improvement of the bound. Both the construction and the proof of our algorithm are very simple; the worst-case complexity of our algorithm is evaluated by an easy argument. We also give some computer experimental results showing the efficiency of our algorithm for concrete parameters, and discuss potential applications of our result to security evaluations of existing factoring-based primitives.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Cheon, J.H., Coron, J.-S., Kim, J., Lee, M.S., Lepoint, T., Tibouchi, M., Yun, A.: Batch fully homomorphic encryption over the integers. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 315–335. Springer, Heidelberg (2013)
Coppersmith, D.: Finding a small root of a bivariate integer equation; factoring with high bits known. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 178–189. Springer, Heidelberg (1996)
Galbraith, S.D.: Mathematics of Public Key Cryptography. Cambridge University Press (2012)
Kurosawa, K., Ueda, T.: How to Factor N \(_{1}\) and N \(_{2}\) When \(p_{1}\) = \(p_{2}\) mod 2\(^t\). In: Sakiyama, K., Terada, M. (eds.) IWSEC 2013. LNCS, vol. 8231, pp. 217–225. Springer, Heidelberg (2013)
Lenstra Jr, H.W.: Factoring Integers with Elliptic Curves. Ann. Math. 126, 649–673 (1987)
Lenstra, A.K., Lenstra Jr, H.W.: The Development of the Number Field Sieve. Springer, Heidelberg (1993)
Lu, Y., Peng, L., Zhang, R., Lin, D.: Towards Optimal Bounds for Implicit Factorization Problem, IACR Cryptology ePrint Archive 2014/825 (2014)
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)
Nuida, K., Kurosawa, K.: (Batch) Fully Homomorphic Encryption over Integers for Non-Binary Message Spaces. In: EUROCRYPT 2015 (2015, to appear). IACR Cryptology ePrint Archive 2014/777 (2014)
Okamoto, T., Uchiyama, S.: A New Public-Key Cryptosystem as Secure as Factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 308–318. Springer, Heidelberg (1998)
Pomerance, C.: The quadratic sieve factoring algorithm. In: Beth, T., Cot, N., Ingemarsson, I. (eds.) EUROCRYPT 1984. LNCS, vol. 209, pp. 169–182. Springer, Heidelberg (1985)
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 Transactions on Information Theory 57(6), 4002–4013 (2011)
Takagi, T.: Fast RSA-type Cryptosystem Modulo \(p^{k}q\). In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, p. 318. Springer, Heidelberg (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Nuida, K., Itakura, N., Kurosawa, K. (2015). A Simple and Improved Algorithm for Integer Factorization with Implicit Hints. In: Nyberg, K. (eds) Topics in Cryptology –- CT-RSA 2015. CT-RSA 2015. Lecture Notes in Computer Science(), vol 9048. Springer, Cham. https://doi.org/10.1007/978-3-319-16715-2_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-16715-2_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16714-5
Online ISBN: 978-3-319-16715-2
eBook Packages: Computer ScienceComputer Science (R0)