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).
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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)
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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
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DOI: https://doi.org/10.1007/978-3-319-18681-8_28
Publisher Name: Springer, Cham
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