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A Strategy for Finding Roots of Multivariate Polynomials with New Applications in Attacking RSA Variants

  • Ellen Jochemsz
  • Alexander May
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4284)

Abstract

We describe a strategy for finding small modular and integer roots of multivariate polynomials using lattice-based Coppersmith techniques. Applying our strategy, we obtain new polynomial-time attacks on two RSA variants. First, we attack the Qiao-Lam scheme that uses a Chinese Remaindering decryption process with a small difference in the private exponents. Second, we attack the so-called Common Prime RSA variant, where the RSA primes are constructed in a way that circumvents the Wiener attack.

Keywords

lattices small roots Coppersmith’s method RSA variants cryptanalysis 

References

  1. 1.
    Boneh, D., Durfee, G.: Cryptanalysis of RSA with Private Key d Less Than N 0.292. IEEE Transactions on Information Theory 46, 1339–1349 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Blömer, J., May, A.: A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 251–267. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Coppersmith, D.: Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities. Journal of Cryptology 10, 233–260 (1997)MATHCrossRefMathSciNetGoogle 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.
    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
  9. 9.
    Hinek, M.J.: Another Look at Small RSA Exponents. In: Pointcheval, D. (ed.) CT-RSA 2006. LNCS, vol. 3860, pp. 82–98. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Hinek, M.J., Stinson, D.R.: An Inequality About Factors of Multivariate Polynomials (2006), http://www.cacr.math.uwaterloo.ca/techreports/2006/cacr2006-15.pdf
  11. 11.
    Howgrave-Graham, N.: Finding Small Roots of Univariate Modular Equations Revisited. In: Darnell, M.J. (ed.) Cryptography and Coding 1997. LNCS, vol. 1355, pp. 131–142. Springer, Heidelberg (1997)Google Scholar
  12. 12.
    Lim, C.H., Lee, P.J.: Security and performance of server-aided RSA computation protocols. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 70–83. Springer, Heidelberg (1995)Google Scholar
  13. 13.
    Lenstra, A., Lenstra Jr., H., Lovász, L.: Factoring Polynomials with Rational Coefficients. Mathematische Ann. 261, 513–534 (1982)Google Scholar
  14. 14.
    May, A.: New RSA Vulnerabilities Using Lattice Reduction Methods, Ph.D Thesis, University of Paderborn (2003)Google Scholar
  15. 15.
    McKee, J., Pinch, R.: Further attacks on server-aided RSA cryptosystems (1998), http://citeseer.ist.psu.edu/388295.html
  16. 16.
    Nguên, P.Q., Stehlé, D.: Floating-Point LLL Revisited. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Qiao, G., Lam, K.-Y.: RSA Signature Algorithm for Microcontroller Implementation. In: Schneier, B., Quisquater, J.-J. (eds.) CARDIS 1998. LNCS, vol. 1820, pp. 353–356. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. 18.
    Shoup, V.: NTL: A Library for doing Number Theory, online available at: http://www.shoup.net/ntl/index.html
  19. 19.
    Wiener, M.: Cryptanalysis of Short RSA Secret Exponents. IEEE Transactions on Information Theory 36, 553–558 (1990)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ellen Jochemsz
    • 1
  • Alexander May
    • 2
  1. 1.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands
  2. 2.Faculty of Computer ScienceTU DarmstadtDarmstadtGermany

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