Abstract
We present three attacks on the Prime Power RSA with modulus N = p r q. In the first attack, we consider a public exponent e satisfying an equation ex − φ(N)y = z where φ(N) = p r − 1(p − 1)(q − 1). We show that one can factor N if the parameters |x| and |z| satisfy \(|xz|<N^\frac{r(r-1)}{(r+1)^2}\) thereby extending the recent results of Sakar [16]. In the second attack, we consider two public exponents e 1 and e 2 and their corresponding private exponents d 1 and d 2. We show that one can factor N when d 1 and d 2 share a suitable amount of their most significant bits, that is \(|d_1-d_2|<N^{\frac{r(r-1)}{(r+1)^2}}\). The third attack enables us to factor two Prime Power RSA moduli \(N_1=p_1^rq_1\) and \(N_2=p_2^rq_2\) when p 1 and p 2 share a suitable amount of their most significant bits, namely, \(|p_1-p_2|<\frac{p_1}{2rq_1q_2}\).
Partially supported by the French SIMPATIC (SIM and PAiring Theory for Information and Communications security).
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Blömer, J., May, A.: A Generalized Wiener Attack on RSA. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 1–13. Springer, Heidelberg (2004)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N 0.292. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 1–11. Springer, Heidelberg (1999)
Boneh, D., Durfee, G., Howgrave-Graham, N.: Factoring tex2html_wrap_inline127 for Large r. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 326–337. Springer, Heidelberg (1999)
Compaq Computer Corperation. Cryptography using Compaq multiprime technology in a parallel processing environment (2002), ftp://ftp.compaq.com/pub/solutions/CompaqMultiPrimeWP.pdf
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. Journal of Cryptology 10(4), 233–260 (1997)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers. Oxford University Press, London (1975)
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)
Hinek, M.J.: Cryptanalysis of RSA and its variants. Chapman & Hall/CRC Cryptography and Network Security. CRC Press, Boca Raton (2010)
Lenstra, H.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)
Lenstra, A.K., Lenstra Jr., H.W.: The Development of the Number Field Sieve. Lecture Notes in Mathematics, vol. 1554. Springer, Heidelberg (1993)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 513–534 (1982)
Lu, Y., Zhang, R., Lin, D.: New Results on Solving Linear Equations Modulo Unknown Divisors and its Applications, Cryptology ePrint Archive, Report 2014/343 (2014), https://eprint.iacr.org/2014/343
May, A.: Secret Exponent Attacks on RSA-type Schemes with Moduli N = p r q. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 218–230. Springer, Heidelberg (2004)
May, A.: Using LLL-reduction for solving RSA and factorization problems: a survey. In: LLL+25 Conference in Honour of the 25th Birthday of the LLL Algorithm. Springer, Heidelberg (2007)
Rivest, R., Shamir, A., Adleman, L.: A Method for Obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21(2), 120–126 (1978)
Sarkar, S.: Small secret exponent attack on RSA variant with modulus N = p r q. Designs, Codes and Cryptography 73(2), 383–392 (2015)
Takagi, T.: Fast RSA-type cryptosystem modulo p k q. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 318–326. Springer, Heidelberg (1998)
Wiener, M.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36, 553–558 (1990)
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
Nitaj, A., Rachidi, T. (2015). New Attacks on RSA with Moduli N = p r q . In: El Hajji, S., Nitaj, A., Carlet, C., Souidi, E. (eds) Codes, Cryptology, and Information Security. C2SI 2015. Lecture Notes in Computer Science(), vol 9084. Springer, Cham. https://doi.org/10.1007/978-3-319-18681-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-319-18681-8_28
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18680-1
Online ISBN: 978-3-319-18681-8
eBook Packages: Computer ScienceComputer Science (R0)