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

  • Jean-Luc Beuchat
  • Emmanuel López-Trejo
  • Luis Martínez-Ramos
  • Shigeo Mitsunari
  • Francisco Rodríguez-Henríquez
Part of the Lecture Notes in Computer Science book series (LNCS, 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 ηT pairing supersingular curve finite field arithmetic multi-core 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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. 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. 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)MATHCrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 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)CrossRefGoogle Scholar
  7. 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)CrossRefMathSciNetGoogle Scholar
  8. 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. 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. 10.
    Fong, K., Hankerson, D., López, J., Menezes, A.: Field inversion and point halving revisited. IEEE Transactions on Computers 53(8), 1047–1059 (2004)CrossRefGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 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. 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)MATHMathSciNetGoogle Scholar
  14. 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. 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)CrossRefGoogle Scholar
  16. 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. 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)MATHMathSciNetGoogle Scholar
  18. 18.
    Hess, F.: Pairing lattices. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 18–38. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Hess, F., Smart, N., Vercauteren, F.: The Eta pairing revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)CrossRefMathSciNetGoogle Scholar
  20. 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. 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)CrossRefGoogle Scholar
  22. 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. 23.
    Miller, V.S.: Short programs for functions on curves (1986), http://crypto.stanford.edu/miller
  24. 24.
    Miller, V.S.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Ó hÉigeartaigh, C.: Pairing Computation on Hyperelliptic Curves of Genus 2. PhD thesis, Dublin City University (2006)Google Scholar
  26. 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. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 29.
    Vercauteren, F.: Optimal pairings. Cryptology ePrint Archive, Report 2008/096 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jean-Luc Beuchat
    • 1
  • Emmanuel López-Trejo
    • 2
  • Luis Martínez-Ramos
    • 3
  • Shigeo Mitsunari
    • 4
  • Francisco Rodríguez-Henríquez
    • 3
  1. 1.Graduate School of Systems and Information EngineeringUniversity of TsukubaIbarakiJapan
  2. 2.Nehalem Platform Validation, Intel Guadalajara Design CenterTlaquepaqueMéxico
  3. 3.Computer Science DepartmentCentro de Investigación y de Estudios Avanzados del IPNMéxico CityMéxico
  4. 4.Cybozu Labs, Inc.Akasaka Twin Tower East 15F, 2-17-22 Akasaka, Minato-kuTokyo

Personalised recommendations