Skip to main content
SpringerLink
Log in

Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree

(Extended Abstract)

  • Conference paper
  • First Online:
Progress in Cryptology — INDOCRYPT 2001 (INDOCRYPT 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2247))

Included in the following conference series:

Abstract

We analyze the Gaudry-Hess-Smart (GHS) Weil descent attack on the elliptic curve discrete logarithm problem (ECDLP)for elliptic curves defined over characteristic two finite fields of composite extension degree. For each such field F2 N, N ∈ [160, 600], we identify elliptic curve parameters such that (i)there should exist a cryptographically interesting elliptic curve E over F2 N with these parameters; and (ii)the GHS attack is more efficient for solving the ECDLP in E(F N2 )than for any other cryptographically interesting elliptic curve over F2 N.

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. E. Artin. “Quadratische Körper im Gebiete der höheren Kongruenzen”, Mathematische Zeitschrift, 19 (1924), 207–246.

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, K. Wildanger, “KANT V4”, J. Symbolic Computation, 24 (1997), 267–283.

    Article  MATH  Google Scholar 

  4. A. Enge, P. Gaudry, “A general framework for subexponential discrete logarithm algorithms”, Acta Arithmetica, to appear.

    Google Scholar 

  5. M. Fouquet, P. Gaudry, R. Harley, “An extension of Satoh’s algorithm and its implementation”, J. Ramanujan Mathematical Society, 15 (2000), 281–318.

    MATH  MathSciNet  Google Scholar 

  6. G. Frey, “How to disguise an elliptic curve (Weil descent)”, Talk at ECC’ 98, Waterloo, 1998.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. S. Galbraith, F. Hess, N. Smart, “Extending the GHS Weil descent attack”, preprint, 2001.

    Google Scholar 

  9. S. Galbraith, N. Smart, “A cryptographic application of Weil descent”, Codes and Cryptography, LNCS 1746, 1999, 191–200.

    Chapter  Google Scholar 

  10. P. Gaudry, “An algorithm for solving the discrete log problem on hyperelliptic curves”, Advances in Cryptology—Eurocrypt 2000, LNCS 1807, 2000, 19–34.

    Chapter  Google Scholar 

  11. P. Gaudry, F. Hess, N. Smart, “Constructive and destructive facets of Weil descent on elliptic curves”, preprint, January 2000.

    Google Scholar 

  12. F. Hess, KASH program for performing the GHS attack, 2000.

    Google Scholar 

  13. Internet Engineering Task Force, The OAKLEY Key Determination Protocol, IETF RFC 2412, November 1998.

    Google Scholar 

  14. M. Jacobson, A. Menezes, A. Stein, “Solving elliptic curve discrete logarithm problems using Weil descent”, J. Ramanujan Mathematical Society, to appear.

    Google Scholar 

  15. A. Joux. Personal communication. June 2001.

    Google Scholar 

  16. A. Joux, R. Lercier, “Improvements on the general number field sieve for discrete logarithms in finite fields”, Math. Comp., to appear.

    Google Scholar 

  17. A. Menezes, T. Okamoto, S. Vanstone, “Reducing elliptic curve logarithms to logarithms in a finite field”, IEEE Trans. Info. Th., 39 (1993), 1639–1646.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Menezes, M. Qu, “Analysis of the Weil descent attack of Gaudry, Hess and Smart”, Topics in Cryptology—CT-RSA 2001, LNCS 2020, 2001, 308–318.

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. S. Paulus, H. Rück, “Real amd imaginary quadratic representations of hyperelliptic function fields”, Math. Comp., 68 (1999), 1233–1241.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Pohlig, M. Hellman, “An improved algorithm for computing logs over GF(p) and its cryptographic significance”, IEEE Trans. Info. Th., 24 (1978), 106–110.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. T. Satoh, “The canonical lift of an ordinary elliptic curve over a finite field and its point counting”, J. Ramanujan Mathematical Society, 15 (2000), 247–270.

    MATH  MathSciNet  Google Scholar 

  24. E. Teske, “Speeding up Pollard’s rho method for computing discrete logarithms”, Algorithmic Number Theory, LNCS 1423, 1998, 541–554.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of C&O, University of Waterloo, Canada

    Markus Maurer, Alfred Menezes & Edlyn Teske

Authors
  1. Markus Maurer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alfred Menezes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edlyn Teske
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology, Madras, Chennai, India

    C. Pandu Rangan

  2. Department of Computer Science, Hong Kong University of Science and Technology, Hong Kong

    Cunsheng Ding

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Maurer, M., Menezes, A., Teske, E. (2001). Analysis of the GHS Weil Descent Attack on the ECDLP over Characteristic Two Finite Fields of Composite Degree. In: Rangan, C.P., Ding, C. (eds) Progress in Cryptology — INDOCRYPT 2001. INDOCRYPT 2001. Lecture Notes in Computer Science, vol 2247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45311-3_19

Download citation

  • DOI: https://doi.org/10.1007/3-540-45311-3_19

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43010-0

  • Online ISBN: 978-3-540-45311-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation