A New Partial Key Exposure Attack on Multi-power RSA

  • Muhammed F. EsginEmail author
  • Mehmet S. Kiraz
  • Osmanbey Uzunkol
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9270)


An important attack on multi-power RSA (\(N=p^rq\)) was introduced by Sarkar in 2014, by extending the small private exponent attack of Boneh and Durfee on classical RSA. In particular, he showed that N can be factored efficiently for \(r=2\) with private exponent d satisfying \(d<N^{0.395}\). In this paper, we generalize this work by introducing a new partial key exposure attack for finding small roots of polynomials using Coppersmith’s algorithm and Gröbner basis computation. Our attack works for all multi-power RSA exponents e (resp. d) when the exponent d (resp. e) has full size bit length. The attack requires prior knowledge of least significant bits (LSBs), and has the property that the required known part of LSB becomes smaller in the size of e. For practical validation of our attack, we demonstrate several computer algebra experiments.


Multi-power RSA Integer factorization Partial key exposure Coppersmith’s method Small roots of polynomials 


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  1. 1.
    Blömer, J., May, A.: New partial key exposure attacks on RSA. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 27–43. Springer, Heidelberg (2003) CrossRefGoogle Scholar
  2. 2.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Boneh, D., Durfee, G., Frankel, Y.: An attack on RSA given a small fraction of the private key bits. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 25–34. Springer, Heidelberg (1998) CrossRefGoogle Scholar
  4. 4.
    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) CrossRefGoogle Scholar
  5. 5.
    Coppersmith, D.: Finding a small root of a univariate modular equation. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996) CrossRefGoogle Scholar
  6. 6.
    Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. Journal of Cryptology 10(4), 233–260 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Coron, J.-S.: Finding small roots of bivariate integer polynomial equations revisited. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 492–505. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  8. 8.
    Coron, J.-S.: Finding small roots of bivariate integer polynomial equations: a direct approach. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 379–394. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  9. 9.
    Ernst, M., Jochemsz, E., May, A., de Weger, B.: Partial key exposure attacks on RSA up to full size exponents. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 371–386. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  10. 10.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner Bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, ISSAC 2002, New York, NY, USA, pp. 75–83. ACM (2002)Google Scholar
  11. 11.
    Howgrave-Graham, N.: Finding small roots of univariate modular equations revisited. In: Darnell, M. (ed.) Crytography and Coding. Lecture Notes in Computer Science, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. 12.
    Huang, Z., Hu, L., Xu, J., Peng, L., Xie, Y.: Partial key exposure attacks on Takagi’s variant of RSA. In: Boureanu, I., Owesarski, P., Vaudenay, S. (eds.) ACNS 2014. LNCS, vol. 8479, pp. 134–150. Springer, Heidelberg (2014) Google Scholar
  13. 13.
    Itoh, K., Kunihiro, N., Kurosawa, K.: Small secret key attack on a variant of RSA (due to Takagi). In: Malkin, T. (ed.) CT-RSA 2008. LNCS, vol. 4964, pp. 387–406. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  14. 14.
    Joye, M., Lepoint, T.: Partial key exposure on RSA with private exponents larger than N. In: Ryan, M.D., Smyth, B., Wang, G. (eds.) ISPEC 2012. LNCS, vol. 7232, pp. 369–380. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  15. 15.
    Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996) Google Scholar
  16. 16.
    Lenstra Jr., A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261(4), 515–534 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    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).
  18. 18.
    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) CrossRefGoogle Scholar
  19. 19.
    Nguyên, P.Q., Stehlé, D.: Floating-Point LLL revisited. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Sarkar, S.: Small secret exponent attack on RSA variant with modulus \(N=p^rq\). Designs, Codes and Cryptography 73(2), 383–392 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Takagi, T.: Fast RSA-type cryptosystem modulo \(p^kq\). In: Krawczyk, H. (ed.) Advances in Cryptology - CRYPTO ’98. Lecture Notes in Computer Science, vol. 1462, pp. 318–326. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  22. 22.
    Wiener, M.J.: Cryptanalysis of short RSA secret exponents. IEEE Transactions on Information Theory 36, 553–558 (1990)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Muhammed F. Esgin
    • 1
    • 2
    Email author
  • Mehmet S. Kiraz
    • 1
  • Osmanbey Uzunkol
    • 1
  1. 1.TÜBİTAK BİLGEM UEKAEKocaeliTurkey
  2. 2.Graduate School of Natural and Applied Sciencesİstanbul Şehir UniversityIstanbulTurkey

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