Skip to main content
SpringerLink
Log in

Efficient Computations of the Tate Pairing for the Large MOV Degrees

  • Conference paper
  • First Online:
Book cover Information Security and Cryptology — ICISC 2002 (ICISC 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2587))

Included in the following conference series:

Abstract

The Tate pairing hasp lenty of attractive applications, e.g., ID-based cryptosystems, short signatures, etc. Recently several fast implementationsof the Tate pairing hasb een reported, which make it appear that the Tate pairing is capable to be used in practical applications. The computation time of the Tate pairing strongly depends on underlying elliptic curves and definition fields. However these fast implementation are restricted to supersingular curves with small MOV degrees. In this paper we propose several improvements of computing the Tate pairing over general elliptic curves over finite fields IF q (q = pm, p > 3) - some of them can be efficiently applied to supersingular curves. The proposed methods can be combined with previous techniques. The proposed algorithm iss pecially effective upon the curvest hat hasa large MOV degree k. We develop several formulas that compute the Tate pairing using the small number of multiplications over IFq k. For k = 6, the proposed algorithm is about 20% faster than previously fastest algorithm.

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. D. Boneh, and M. Franklin, “Identity-based encryption from the Weil pairing”, CRYPTO 2001, LNCS 2139, pp.213–229, Springer-Verlag, 2001. 284

    Google Scholar 

  2. P. Barreto, H. Kim, B. Lynn, and M. Scott, “Efficient Algorithmsfor Pairing-Based Cryptosystems”, CRYPTO 2002, LNCS 2442, pp.354–368, Springer-Verlag, 2002. 284, 292, 293, 294

    Google Scholar 

  3. D. Coppersmith, “Evaluating Logarithms in GF(2 n)”, STOC 1984, pp.201–207, 1983. 284

    Google Scholar 

  4. H. Cohen, A. Miyaji and T. Ono, “Efficient elliptic curve exponentiation using mixed coordinates”, Asiacrypt’98, LNCS 1514, pp.51–65, Springer-Verlag, 1998. 285

    Google Scholar 

  5. R. Dunport, A. Enge, and F. Morain, “Building curveswit h arbitrary small MOV degree over finite prime fields”, Cryptology ePrint Archive, Report 2002/094, 2002. 284

    Google Scholar 

  6. G. Frey, M. Müller, and H. Rück, “The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems”, IEEE Trans. on Information Theory, vol.45, pp.1717–1718, 1999. 283

    Article  MATH  Google Scholar 

  7. S.D. Galbraith, “Supersingular Curves in Cryptography”, Asiacrypt 2001, LNCS 2248, pp.495–513, Springer-Verlag, 2001. 284

    Chapter  Google Scholar 

  8. S.D. Galbraith, K. Harrison, and D. Soldera, “Implementing the Tate pairing”, ANTS V, LNCS 2369, pp.324–337, Springer-Verlag, 2002. 284, 287, 291, 292, 294

    Google Scholar 

  9. F. Hess, “Exponent Group Signature Schemes and Efficient Identity Based Signature Schemes Based on Pairings”, Cryptology ePrint Archive, Report 2002/012, 2002. 284

    Google Scholar 

  10. K. Itoh, M. Takenaka, N. Torii, S. Temma, and Y. Kurihara, “Fast Implementation of Public-Key Cryptography on DSP TMS320C6201”, CHES’99, LNCS 1717, pp.61–72, 1999. 291

    Google Scholar 

  11. A. Joux, “A One Round Protocol for Tripartite Diffie-Hellman”, ANTS IV, LNCS 1838, pp.385–393, Springer-Verlag, 2000. 284

    Google Scholar 

  12. A. Joux, “The Weil and Tate Pairings asBu ilding Blocksfor Public Key Cryptosystems (survey)”, ANTS V, LNCS 2369, pp.20–32, Springer-Verlag, 2002. 284, 294

    Google Scholar 

  13. N. Koblitz, “Elliptic curve cryptosystems”,Math. of Comp., vol.48, pp.203–209, 1987. 283

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Menezes, “Elliptic Curve Public Key Cryptosystems”, Kluwer Academic Publishers, 1993. 288, 293

    Google Scholar 

  15. V. Miller, “Use of elliptic curves in cryptography”, CRYPTO’85, LNCS 218. p.417–426, Springer-Verlag, 1986. 283

    Google Scholar 

  16. A. Miyaji, M. Nakabayashi, and S. Takano, “New explicit conditions of elliptic curve tracesfor FR-reduction”, IEICE Trans. Fundamentals, E84-A(5), May, 2001. 284

    Google Scholar 

  17. A. Menezes, T. Okamoto, and S. Vanstone, “Reducing Elliptic Curve Logarithmst o Logarithms in a Finite Field”, IEEE Trans. on Information Theory, vol.39, pp.1639–1646, 1993. 283, 287

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Okamoto, P. Pointcheval, “The Gap Problems: a new class of problems for the security of cryptographic primitives”, PKC 2001, LNCS 1992, pp.104–118, Springer-Verlag, 2001. 284

    Google Scholar 

  19. K. G. Paterson, “ID-based Signatures from Pairings on Elliptic Curves”, Cryptology ePrint Archive, Report 2002/004, 2002. 284

    Google Scholar 

  20. D. Page, and N. Smart, “Hardware Implementation of Finite Fieldsof Characteristic Three”, to appear in the proceedings of CHES 2002. 284

    Google Scholar 

  21. N.P. Smart, “An Identity Based Authentificated Key Agreement Protocol Based on the Weil Pairing”, Cryptology ePrint Archive, Report 2001/111, 2001. 284

    Google Scholar 

  22. R. Sakai, K. Ohgishi, and M. Kasahara, “Cryptosystems Based on Pairing”, 2000 Symposium on Cryptography and Information Security (SCIS 2000), Okinawa, Japan, Jan. 26-28, 2000. 284

    Google Scholar 

  23. N.P. Smart, and J. Westwood, “Point Multiplication on Ordinary Elliptic Curveso ver Fieldsof Characteristic Three”, Cryptology ePrint Archive, Report 2002/114, 2002. 284

    Google Scholar 

  24. E.R. Verheul, “Self-Blindable Credential Certificatesfrom the Weil pairing”, ASIACRYPT 2001, LNCS 2248, pp.533–551, Springer-Verlag, 2001. 284

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FUJITSU LABORATORIES Ltd., 4-1-1, Kamikodanaka, Nakahara-ku, 211-8588, Kawasaki, Japan

    Tetsuya Izu

  2. Fachbereich Informatik, Technische Universität Darmstadt, Alexanderstr.10, D-64283, Darmstadt, Germany

    Tsuyoshi Takagi

Authors
  1. Tetsuya Izu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsuyoshi Takagi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Pohnag University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang, 790-784, Kyungbuk, Korea

    Pil Joong Lee

  2. Sejong University, 98, Gunja-dong, Gwangjin-gu, 143-747, Seoul, Korea

    Chae Hoon Lim

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Izu, T., Takagi, T. (2003). Efficient Computations of the Tate Pairing for the Large MOV Degrees. In: Lee, P.J., Lim, C.H. (eds) Information Security and Cryptology — ICISC 2002. ICISC 2002. Lecture Notes in Computer Science, vol 2587. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36552-4_20

Download citation

  • DOI: https://doi.org/10.1007/3-540-36552-4_20

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00716-6

  • Online ISBN: 978-3-540-36552-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation