Skip to main content
Book cover

International Conference on Cryptology and Network Security

CANS 2009: Cryptology and Network Security pp 413–432Cite as

Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 5888)

Abstract

This paper describes the design of a fast multi-core library for the cryptographic Tate pairing over supersingular elliptic curves. For the computation of the reduced modified Tate pairing over \(\mathbb{F}_{3^{509}}\), we report calculation times of just 2.94 ms and 1.87 ms on the Intel Core2 and Intel Core i7 architectures, respectively. We also try to answer one important design question that arises: how many cores should be utilized for a given application?

Keywords

  • Tate pairing
  • η T pairing
  • supersingular curve
  • finite field arithmetic
  • multi-core

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-10433-6_28
  • Chapter length: 20 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-10433-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   149.00
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmadi, O., Rodríguez-Henríquez, F.: Low complexity cubing and cube root computation over \(\mathbb{F}_{3^m}\) in standard basis. Cryptology ePrint Archive, Report 2009/070 (2009)

    Google Scholar 

  2. Barreto, P.S.L.M.: A note on efficient computation of cube roots in characteristic 3. Cryptology ePrint Archive, Report 2004/305 (2004)

    Google Scholar 

  3. Barreto, P.S.L.M., Galbraith, S.D., Ó hÉigeartaigh, C., Scott, M.: Efficient pairing computation on supersingular Abelian varieties. Designs, Codes and Cryptography 42, 239–271 (2007)

    MATH  CrossRef  Google Scholar 

  4. Barreto, P.S.L.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient algorithms for pairing-based cryptosystems. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 354–368. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  5. Barreto, P.S.L.M., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Preneel, B., Tavares, S. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  6. Beuchat, J.-L., Brisebarre, N., Detrey, J., Okamoto, E., Rodríguez-Henríquez, F.: A comparison between hardware accelerators for the modified tate pairing over \(\mathbb{F}_{2^m}\) and \(\mathbb{F}_{3^m}\). In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 297–315. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  7. Beuchat, J.-L., Brisebarre, N., Detrey, J., Okamoto, E., Shirase, M., Takagi, T.: Algorithms and arithmetic operators for computing the η T pairing in characteristic three. IEEE Transactions on Computers 57(11), 1454–1468 (2008)

    CrossRef  MathSciNet  Google Scholar 

  8. Beuchat, J.-L., Detrey, J., Estibals, N., Okamoto, E., Rodríguez-Henríquez, F.: Hardware accelerator for the Tate pairing in characteristic three based on Karatsuba–Ofman multipliers. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 225–239. Springer, Heidelberg (2009)

    Google Scholar 

  9. Duursma, I., Lee, H.S.: Tate pairing implementation for hyperelliptic curves y 2 = x p − x + d. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 111–123. Springer, Heidelberg (2003)

    Google Scholar 

  10. Fong, K., Hankerson, D., López, J., Menezes, A.: Field inversion and point halving revisited. IEEE Transactions on Computers 53(8), 1047–1059 (2004)

    CrossRef  Google Scholar 

  11. Galbraith, S.D., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 324–337. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  12. Grabher, P., Großschädl, J., Page, D.: On software parallel implementation of cryptographic pairings. In: SAC 2008. LNCS, vol. 5381, pp. 34–49. Springer, Heidelberg (2008)

    Google Scholar 

  13. Granger, R., Page, D., Stam, M.: On small characteristic algebraic tori in pairing-based cryptography. LMS Journal of Computation and Mathematics 9, 64–85 (2006)

    MATH  MathSciNet  Google Scholar 

  14. Gueron, S., Kounavis, M.E.: Carry-less multiplication and its usage for computing the GCM mode. Intel Corporation White Paper (May 2009)

    Google Scholar 

  15. Hankerson, D., López Hernandez, J., Menezes, A.J.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  16. Hankerson, D., Menezes, A., Scott, M.: Software Implementation of Pairings. Cryptology and Information Security Series, ch. 12, pp. 188–206. IOS Press, Amsterdam (2009)

    Google Scholar 

  17. Harrison, K., Page, D., Smart, N.P.: Software implementation of finite fields of characteristic three, for use in pairing-based cryptosystems. LMS Journal of Computation and Mathematics 5, 181–193 (2002)

    MATH  MathSciNet  Google Scholar 

  18. Hess, F.: Pairing lattices. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 18–38. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  19. Hess, F., Smart, N., Vercauteren, F.: The Eta pairing revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)

    CrossRef  MathSciNet  Google Scholar 

  20. Kammler, D., Zhang, D., Schwabe, P., Scharwaechter, H., Langenberg, M., Auras, D., Ascheid, G., Leupers, R., Mathar, R., Meyr, H.: Designing an ASIP for cryptographic pairings over Barreto-Naehrig curves. Cryptology ePrint Archive, Report 2009/056 (2009)

    Google Scholar 

  21. Kawahara, Y., Aoki, K., Takagi, T.: Faster implementation of η T pairing over GF(3m) using minimum number of logical instructions for GF(3)-addition. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 282–296. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

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

    Google Scholar 

  23. Miller, V.S.: Short programs for functions on curves (1986), http://crypto.stanford.edu/miller

  24. Miller, V.S.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)

    MATH  CrossRef  MathSciNet  Google Scholar 

  25. Ó hÉigeartaigh, C.: Pairing Computation on Hyperelliptic Curves of Genus 2. PhD thesis, Dublin City University (2006)

    Google Scholar 

  26. Schroeppel, R., Orman, H., O’Malley, S.W., Spatscheck, O.: Fast key exchange with elliptic curve systems. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 43–56. Springer, Heidelberg (1995)

    Google Scholar 

  27. Shirase, M., Takagi, T., Choi, D., Han, D., Kim, H.: Efficient computation of Eta pairing over binary field with Vandermonde matrix. ETRI Journal 31(2), 129–139 (2009)

    CrossRef  Google Scholar 

  28. Shu, C., Kwon, S., Gaj, K.: Reconfigurable computing approach for Tate pairing cryptosystems over binary fields. IEEE Transactions on Computers 58(9), 1221–1237 (2009)

    CrossRef  Google Scholar 

  29. Vercauteren, F.: Optimal pairings. Cryptology ePrint Archive, Report 2008/096 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8573, Japan

    Jean-Luc Beuchat

  2. Nehalem Platform Validation, Intel Guadalajara Design Center, Periférico Sur 7980 Edificio 4E, 45600, Tlaquepaque, Jalisco, México

    Emmanuel López-Trejo

  3. Computer Science Department, Centro de Investigación y de Estudios Avanzados del IPN, Av. Instituto Politécnico Nacional No. 2508, 07300, México City, México

    Luis Martínez-Ramos & Francisco Rodríguez-Henríquez

  4. Cybozu Labs, Inc., Akasaka Twin Tower East 15F, 2-17-22 Akasaka, Minato-ku, Tokyo, 107-0052

    Shigeo Mitsunari

Authors
  1. Jean-Luc Beuchat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emmanuel López-Trejo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luis Martínez-Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shigeo Mitsunari
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Francisco Rodríguez-Henríquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. AT&T Labs Research, Florham Park, NJ, USA

    Juan A. Garay

  2. Japan Advanced Institute of Science and Technology (JAIST), Nomi, Ishikawa, Japan

    Atsuko Miyaji

  3. National Institute of Advanced Industrial Science and Technoloy (AIST), Tokyo, Japan

    Akira Otsuka

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Beuchat, JL., López-Trejo, E., Martínez-Ramos, L., Mitsunari, S., Rodríguez-Henríquez, F. (2009). Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves. In: Garay, J.A., Miyaji, A., Otsuka, A. (eds) Cryptology and Network Security. CANS 2009. Lecture Notes in Computer Science, vol 5888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10433-6_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10433-6_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10432-9

  • Online ISBN: 978-3-642-10433-6

  • eBook Packages: Computer ScienceComputer Science (R0)