Using Bleichenbacher”s Solution to the Hidden Number Problem to Attack Nonce Leaks in 384-Bit ECDSA

  • Elke De Mulder
  • Michael Hutter
  • Mark E. Marson
  • Peter Pearson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8086)

Abstract

In this paper we describe an attack against nonce leaks in 384-bit ECDSA using an FFT-based attack due to Bleichenbacher. The signatures were computed by a modern smart card. We extracted the low-order bits of each nonce using a template-based power analysis attack against the modular inversion of the nonce. We also developed a BKZ-based method for the range reduction phase of the attack, as it was impractical to collect enough signatures for the collision searches originally used by Bleichenbacher. We confirmed our attack by extracting the entire signing key using a 5-bit nonce leak from 4000 signatures.

Keywords

Side Channel Analysis ECDSA Modular Inversion Hidden Number Problem Bleichenbacher FFT LLL BKZ 

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References

  1. 1.
    Minutes from the IEEE P1363 Working Group for Public-Key Cryptography Standards (November 15, 2000)Google Scholar
  2. 2.
    ANSI X9.62:2005: Public Key Cryptography for the Financial Services Industry. In: The Elliptic Curve Digital Signature Algorithm, ECDSA (2005)Google Scholar
  3. 3.
    Babai, L.: On Lovász’ Lattice Reduction and the Nearest Lattice Point Problem. Combinatorica 6(1), 1–13 (1986)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bleichenbacher, D.: On The Generation of One-Time Keys in DL Signature Schemes. Presentation at IEEE P1363 Working Group meeting (November 2000)Google Scholar
  5. 5.
    Bleichenbacher, D.: On the Generation of DSA One-Time Keys. Presentation at Cryptography Research, Inc., San Francisco (2007)Google Scholar
  6. 6.
    Boneh, D., Venkatesan, R.: Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 129–142. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Cadé, D., Pujol, X., Stehlé, D.: fplll-4.0.1 Lattice Reduction Library (2012)Google Scholar
  8. 8.
    Chari, S., Rao, J.R., Rohatgi, P.: Template Attacks. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 13–28. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Chen, Y., Nguyen, P.Q.: BKZ 2.0: Better Lattice Security Estimates. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 1–20. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Hachez, G., Quisquater, J.-J.: Montgomery Exponentiation with no Final Subtractions: Improved Results. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 293–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Hamburg, M.: Fast and Compact Elliptic-Curve Cryptography. IACR Cryptology ePrint Archive, 309 (2012)Google Scholar
  12. 12.
    Hedabou, M., Pinel, P., Bènèteau, L.: A Comb Method to Render ECC Resistant Against Side Channel Attacks. IACR Cryptology ePrint Archive, 342 (2004)Google Scholar
  13. 13.
    Howgrave-Graham, N., Smart, N.P.: Lattice Attacks on Digital Signature Schemes. Designs, Codes and Cryptography 23(3), 283–290 (2001)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Hutter, M., Medwed, M., Hein, D., Wolkerstorfer, J.: Attacking ECDSA-Enabled RFID Devices. In: Abdalla, M., Pointcheval, D., Fouque, P.-A., Vergnaud, D. (eds.) ACNS 2009. LNCS, vol. 5536, pp. 519–534. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Joye, M., Tunstall, M.: Exponent Recoding and Regular Exponentiation Algorithms. In: Preneel, B. (ed.) AFRICACRYPT 2009. LNCS, vol. 5580, pp. 334–349. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    Kocher, P.C., Jaffe, J., Jun, B.: Differential Power Analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Lenstra, A.K., Lenstra, H., Lovász, L.: Factoring Polynomials with Rational Coefficients. Mathematische Annalen 261, 515–534 (1982)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Liu, M., Nguyen, P.Q.: Solving BDD by Enumeration: An Update. In: Dawson, E. (ed.) CT-RSA 2013. LNCS, vol. 7779, pp. 293–309. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  20. 20.
    Lochter, M., Merkle, J.: Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation. In: RFC 5639 (Informational) (March 2010)Google Scholar
  21. 21.
    Naccache, D., Nguyen, P.Q., Tunstall, M., Whelan, C.: Experimenting with Faults, Lattices and the DSA. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 16–28. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  22. 22.
    National Institute of Standards and Technology (NIST). FIPS-186-2 (+Change Notice): Digital Signature Standard (DSS) (January 2000), http://www.itl.nist.gov/fipspubs/
  23. 23.
    Nguyen, P.Q., Shparlinski, I.: The Insecurity of the Digital Signature Algorithm with Partially Known Nonces. J. Cryptology 15(3), 151–176 (2002)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Nguyen, P.Q., Shparlinski, I.: The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces. Des. Codes Cryptography 30(2), 201–217 (2003)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Quisquater, J.-J., Koeune, F.: DSA Security Evaluation of the Signature Scheme and Primitive. Technical report, Math RiZK, K2Crypt (February 2002)Google Scholar
  26. 26.
    Schnorr, C.-P., Euchner, M.: Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems. Mathematical Programming 66, 181–199 (1994)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Shoup, V.: NTL: A Library for doing Number Theory (2012)Google Scholar
  28. 28.
    Vaudenay, S.: Evaluation Report on DSA. IPA Work Delivery 1002 (2001)Google Scholar
  29. 29.
    Walter, C.D.: Montgomery Exponentiation needs no Final Subtractions. Electronics Letters 35, 1831–1832 (1999)CrossRefGoogle Scholar
  30. 30.
    Walter, C.D., Thompson, S.: Distinguishing Exponent Digits by Observing Modular Subtractions. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 192–207. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Elke De Mulder
    • 1
  • Michael Hutter
    • 1
    • 2
  • Mark E. Marson
    • 1
  • Peter Pearson
    • 1
  1. 1.Cryptography Research, Inc.San FranciscoUSA
  2. 2.Institute for Applied Information Processing and Communications (IAIK)Graz University of TechnologyGrazAustria

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