Skip to main content
SpringerLink
Log in

Weak Fields for ECC

  • Conference paper
Topics in Cryptology – CT-RSA 2004 (CT-RSA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2964))

Included in the following conference series:

Abstract

We demonstrate that some finite fields, including \(\mathbb{F}_{{2}^{210}}\), are weak for elliptic curve cryptography in the sense that any instance of the elliptic curve discrete logarithm problem for any elliptic curve over these fields can be solved in significantly less time than it takes Pollard’s rho method to solve the hardest instances. We discuss the implications of our observations to elliptic curve cryptography, and list some open problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adleman, L., DeMarrais, J., Huang, M.: A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields. In: Huang, M.-D.A., Adleman, L.M. (eds.) ANTS 1994. LNCS, vol. 877, pp. 28–40. Springer, Heidelberg (1994)

    Google Scholar 

  2. Barrett, P.: Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 311–323. Springer, Heidelberg (1987)

    Google Scholar 

  3. Bernstein, D.: Circuits for integer factorization: A proposal, preprint (2001)

    Google Scholar 

  4. Cantor, D.: Computing in the jacobian of a hyperelliptic curve. Math. Comp. 48, 95–101 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cantor, D., Zassenhaus, H.: A new algorithm for factoring polynomials over finite fields. Math. Comp. 36, 587–592 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  6. Coppersmith, D., Odlyzko, A., Schroeppel, R.: Discrete logarithms in GF(p). Algorithmica 1, 1–15 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Enge, A., Gaudry, P.: A general framework for subexponential discrete logarithm algorithms. Acta Arithmetica 102, 83–103 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. FIPS 186-2, Digital signature standard (DSS), Federal Information Processing Standards Publication 186–2, National Institute of Standards and Technology (2000)

    Google Scholar 

  9. Flassenberg, R., Paulus, S.: Sieving in function fields. Experimental Mathematics 8, 339–349 (1999)

    MATH  MathSciNet  Google Scholar 

  10. Fouquet, M.: Anneau d’endomorphismes et cardinalité des courbes elliptiques: aspects algorithmiques, PhD thesis, École polytechnique, Palaiseau Cedex (2001)

    Google Scholar 

  11. Frey, G.: Applications of arithmetical geometry to cryptographic constructions. In: Proceedings of the Fifth International Conference on Finite Fields and Applications, Springer-Verlag, pp. 128–161. Springer, Heidelberg (2001)

    Google Scholar 

  12. Frey, G., Rück, H.: A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math. Comp. 62, 865–874 (1994)

    MATH  MathSciNet  Google Scholar 

  13. Galbraith, S.: Constructing isogenies between elliptic curves over finite fields. LMS Journal of Computation and Mathematics 2, 118–138 (1999)

    MATH  MathSciNet  Google Scholar 

  14. Galbraith, S.: Weil descent of jacobians. Discrete Applied Mathematics 12, 165–180 (2003)

    Article  MathSciNet  Google Scholar 

  15. Galbraith, S., Hess, F., Smart, N.: Extending the GHS Weil descent attack. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 29–44. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  16. Gallant, R., Lambert, R., Vanstone, S.: Improving the parallelized Pollard lambda search on anomalous binary curves. Math. Comp. 69, 1699–1705 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gaudry, P.: An algorithm for solving the discrete log problem in hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  18. Gaudry, P., Hess, F., Smart, N.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptology 15, 19–46 (2002)

    Article  MathSciNet  Google Scholar 

  19. Hankerson, D.: personal communication (2003)

    Google Scholar 

  20. Hess, F.: The GHS attack revisited. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 374–387. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  21. Hess, F.: personal communication (2003)

    Google Scholar 

  22. Jacobson, M.: personal communication (2003)

    Google Scholar 

  23. Jacobson, M., Menezes, A., Stein, A.: Solving elliptic curve discrete logarithm problems using Weil descent. Journal of the Ramanujan Mathematical Society 16, 231–260 (2001)

    MATH  MathSciNet  Google Scholar 

  24. Jacobson, M., van der Poorten, A.: Computational aspects of NUCOMP. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 120–133. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  25. Kohel, D.: Endomorphism rings of elliptic curves over finite fields, PhD thesis, University of California, Berkeley (1996)

    Google Scholar 

  26. Kuhn, F., Struik, R.: Random walks revisited: Extensions of Pollard’s rho algorithm for computing multiple discrete logarithms. In: Vaudenay, S., Youssef, A.M. (eds.) SAC 2001. LNCS, vol. 2259, pp. 212–229. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  27. López, J., Dahab, R.: High-speed software multiplication in \(\mathbb{F}\) 2 m. In: Roy, B., Okamoto, E. (eds.) INDOCRYPT 2000. LNCS, vol. 1977, pp. 203–212. Springer, Heidelberg (2000)

    Google Scholar 

  28. Maurer, M., Menezes, A., Teske, E.: Analysis of the GHS Weil descent attack on the ECDLP over characteristic two finite fields of composite degree. LMS Journal of Computation and Mathematics 5, 127–174 (2002)

    MATH  MathSciNet  Google Scholar 

  29. Menezes, A., Okamoto, T., Vanstone, S.: Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory 39, 1639–1646 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  30. Menezes, A., Qu, M.: Analysis of the Weil descent attack of Gaudry, Hess and Smart. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, pp. 308–318. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  31. van Oorschot, P., Wiener, M.: Parallel collision search with cryptanalytic applications. J. Cryptology 12, 1–28 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Orman, H.: The OAKLEY key determination protocol. RFC 2412 (1998), Available from http://www.ietf.org

  33. Paulus, S., Stein, A.: Comparing real and imaginary arithmetics for divisor class groups of hyperelliptic curves. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 576–591. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  34. Pollard, J.: Monte Carlo methods for index computation mod p. Math. Comp. 32, 918–924 (1978)

    MATH  MathSciNet  Google Scholar 

  35. Satoh, T., Araki, K.: Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves. Commentarii Mathematici Universitatis Sancti Pauli 47, 81–92 (1998)

    MATH  MathSciNet  Google Scholar 

  36. Semaev, I.: Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p. Math. Comp. 67, 353–356 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  37. Shoup, V.: NTL: A library for doing Number Theory, Available from http://shoup.net/ntl

  38. Smart, N.: The discrete logarithm problem on elliptic curves of trace one. J. Cryptology 12, 193–196 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  39. Smart, N.P.: How secure are elliptic curves over composite extension fields? In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 30–39. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  40. Solinas, J.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  41. Teske, E.: Speeding up Pollard’s rho method for computing discrete logarithms. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 541–554. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  42. Teske, E.: On random walks for Pollard’s rho method. Math. Comp. 70, 809–825 (2000)

    Article  MathSciNet  Google Scholar 

  43. Teske, E.: An elliptic curve trapdoor system, Cryptology ePrint Archive Report 2003/058 (2003)

    Google Scholar 

  44. Wiener, M.: The full cost of cryptanalytic attacks. J. Cryptology (to appear)

    Google Scholar 

  45. Wiener, M., Zuccherato, R.: Faster attacks on elliptic curve cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Waterloo, Canada

    Alfred Menezes & Edlyn Teske

  2. University of Essen, Germany

    Annegret Weng

Authors
  1. Alfred Menezes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edlyn Teske
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Annegret Weng
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. NTT, 3-9-11 Midori-cho, Musashino-shi, 180-8585, Tokyo, Japan

    Tatsuaki Okamoto

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Menezes, A., Teske, E., Weng, A. (2004). Weak Fields for ECC. In: Okamoto, T. (eds) Topics in Cryptology – CT-RSA 2004. CT-RSA 2004. Lecture Notes in Computer Science, vol 2964. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24660-2_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24660-2_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-20996-6

  • Online ISBN: 978-3-540-24660-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation