Skip to main content
SpringerLink
Log in

Efficient Pairing Computation on Elliptic Curves in Hessian Form

  • Conference paper
Information Security and Cryptology - ICISC 2010 (ICISC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 6829))

Included in the following conference series:

Abstract

Pairings in elliptic curve cryptography are functions which map a pair of elliptic curve points to a non-zero element of a finite field. In recent years, many useful cryptographic protocols based on pairings have been proposed. The fast implementations of pairings have become a subject of active research areas in cryptology.

In this paper, we give the geometric interpretation of the group law on Hessian curves. Furthermore, we propose the first algorithm for computing the Tate pairing on elliptic curves in Hessian form. Analysis indicates that it is faster than all algorithms of Tate pairing computation known so far for Weierstrass and Edwards curves excepted for the very special elliptic curves with a 4 = 0, a 6 = b 2.

This work was supported by Specialized Research Fund for the Doctoral Program of Higher Education (No. 200802480019) and National Natural Science Foundation of China (No. 61073150).

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. Arène, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster Computation of Tate Pairings, Cryptology ePrint Archive, Report 2009/155, http://eprint.iacr.org/2009/155.pdf

  2. Bernstein, D.J., Lange, T.: Analysis and optimization of elliptic-curve single-scalar multiplication. In: Mullen, G.L., Panario, D., Shparlinski, I.E. (eds.) Finite fields and applications, Contemp. Math., vol. 461, pp. 1–19. American Mathematical Society, Providence (2008)

    Chapter  Google Scholar 

  3. Boneh, D., Franklin, M.K.: Identity-based encryption from the Weil pairing. SIAM J. Comput. 32(3), 586–615 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boneh, D., Shacham, H.: Group signatures with verifier-local revocation. In: Atluri, V., Pfitzmann, B., McDaniel, P. (eds.) ACM CCS 2004, pp. 168–177. ACM Press, New York (2004)

    Google Scholar 

  5. Chatterjee, S., Sarkar, P., Barua, R.: Efficient computation of Tate pairing in projective coordinate over general characteristic fields. In: Park, C., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 168–181. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  6. Cheng, Z., Nistazakis, M.: Implementing pairing-based cryptosystems. In: 3rd International Workshop on Wireless Security Technologies IWWST 2005, London, UK (April 2005)

    Google Scholar 

  7. Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Adv. Appl. Math. 7(4), 385–434 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Costello, C., Hisil, H., Boyd, C., Nieto, J.M.G., Wong, K.K.H.: Faster pairings on special weierstrass curves. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 89–101. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: Srinathan, K., Rangan, C.P., Yung, M. (eds.) INDOCRYPT 2007. LNCS, vol. 4859, pp. 138–151. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  11. Ionica, S., Joux, A.: Another approach to pairing computation in edwards coordinates. In: Chowdhury, D.R., Rijmen, V., Das, A. (eds.) INDOCRYPT 2008. LNCS, vol. 5365, pp. 400–413. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Joux, A.: The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002, Part V. LNCS, vol. 2369, pp. 18–20. Springer, Heidelberg (2002)

    Google Scholar 

  13. Joux, A.: A one round protocol for tripartite Diffie-Hellman. J. Cryptol. 17(4), 263–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Joye, M., Quisquater, J.-J.: Hessian elliptic curves and side-channel attacks. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  15. Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  16. Miller, V.S.: Short Programs for Functions on Curves, IBM Watson, T.J. Research Center (1986), http://crypto.stanford.edu/miller/miller.ps

  17. Miller, V.S.: The Weil pairing and its efficient calculation. J. Cryptol. 17(4), 235–261 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Smart, N.P.: The Hessian form of an elliptic curve. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 118–125. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and Engineering Department, Shanghai Jiao Tong University, 200240, Shanghai, P.R. China

    Haihua Gu & WenLu Xie

  2. Shanghai Huahong Integrated Circuit Co.,Ltd., 201203, Shanghai, P.R. China

    Haihua Gu & Dawu Gu

Authors
  1. Haihua Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dawu Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. WenLu Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of IT Convergence Application Engineering, Pukyong National University, 599-1 Daeyeon 3-Dong Namgu, 608-737, Busan, Republic of Korea

    Kyung-Hyune Rhee

  2. Department of Computer Science and Information Technology, INHA University, 253 Yonghyun-dong, Nam-gu, 402-751, Incheon, Republic of Korea

    DaeHun Nyang

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gu, H., Gu, D., Xie, W. (2011). Efficient Pairing Computation on Elliptic Curves in Hessian Form. In: Rhee, KH., Nyang, D. (eds) Information Security and Cryptology - ICISC 2010. ICISC 2010. Lecture Notes in Computer Science, vol 6829. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24209-0_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-24209-0_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-24208-3

  • Online ISBN: 978-3-642-24209-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation