Skip to main content
SpringerLink
Log in

Interpolation of the Elliptic Curve Diffie-Hellman Mapping

  • Conference paper
  • First Online:
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2643))

Abstract

We prove lower bounds on the degree of polynomials interpolating the Diffie-Hellman mapping for elliptic curves over finite fields and some related mappings including the discrete logarithm. Our results support the assumption that the elliptic curve Diffe-Hellman key exchange and related cryptosystems are secure.

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. Bach and J.O. Shallit. Algorithmic number theory, Vol.1: Efficient algorithms. MIT Press, Cambridge, 1996.

    MATH  Google Scholar 

  2. I.F. Blake, G. Seroussi, and N.P. Smart. Elliptic curves in cryptography, volume 265 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.

    Google Scholar 

  3. J. Buchmann and D. Weber. Discrete logarithms: recent progress. In Coding theory, cryptography and related areas (Guanajuato, (1998), pages 42–56, Berlin, 2000. Springer.

    Google Scholar 

  4. D. Coppersmith and I.E. Shparlinski. On polynomial approximation of the discrete logarithm and the Diffie-Hellman mapping. J. Cryptology, 13:339–360, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Koblitz. Elliptic cryptosystems. Math. Comp., 48:203–209, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Lang. Elliptic curves: Diophantine analysis. Springer, Berlin, 1978.

    MATH  Google Scholar 

  7. T. Lange and A. Winterhof. Polynomial Interpolation of the Elliptic Curve and XTR Discrete Logarithm. In 8th Annual International Computing and Combinatorics Conference (COCOON’02) (Singapore, 2002), volume 2387 of Lect. Notes Comp. Sci., pages 137–143, Berlin, 2002. Springer.

    Google Scholar 

  8. U.M. Maurer and S. Wolf. The Relationship Between Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms. SIAM Journal on Computing, 28(5):1689–1721, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  9. U.M. Maurer and S. Wolf. The Diffie-Hellman Protocol. Designs, Codes, and Cryptography, 19: 147–171, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  10. A.J. Menezes, T. Okamoto, and S.A. Vanstone. Reducing ellipitc curve logarithms to a finite field. IEEE Trans. on Inform. Theory, 39:1639–1646, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  11. A.J. Menezes, P.C. van Oorschot, and S.A. Vanstone. Handbook of Applied Cryptography. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, 1996.

    Google Scholar 

  12. V. Miller. Use of elliptic curves in cryptography. In Advances in Cryptology — Crypto’ 85, volume 263 of Lect. Notes Comput. Sci., pages 417–426, Berlin, 1986. Springer.

    Google Scholar 

  13. S. Pohlig and M. Hellman. An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. on Inform. Theory, 24:106–110, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44:483–494, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  15. I.E. Shparlinski. Number theoretic methods in cryptography. Complexity lower bounds. Birkhäuser, Basel, 1999.

    MATH  Google Scholar 

  16. J.H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate texts in mathematics. Springer, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Information Security and Cryptology, Ruhr-University of Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Tanja Lange

  2. Temasek Laboratories, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260, Republic of Singapore

    Arne Winterhof

Authors
  1. Tanja Lange
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arne Winterhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, 2540 Dole Street, Holmes Hall 455, Honolulu, HI, 96822, USA

    Marc Fossorier

  2. Department of Mathematics, The Technical University of Denmark, Bldg. 303, DK-2800, Lyngby, Denmark

    Tom Høholdt

  3. IRIT - Université Paul Sabatier, 31602, Toulouse cédex, France

    Alain Poli

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lange, T., Winterhof, A. (2003). Interpolation of the Elliptic Curve Diffie-Hellman Mapping. In: Fossorier, M., Høholdt, T., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2003. Lecture Notes in Computer Science, vol 2643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44828-4_7

Download citation

  • DOI: https://doi.org/10.1007/3-540-44828-4_7

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40111-7

  • Online ISBN: 978-3-540-44828-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation