Delaying Mismatched Field Multiplications in Pairing Computations

  • Craig Costello
  • Colin Boyd
  • Juan Manuel Gonzalez Nieto
  • Kenneth Koon-Ho Wong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6087)


Miller’s algorithm for computing pairings involves performing multiplications between elements that belong to different finite fields. Namely, elements in the full extension field \(\mathbb{F}_{p^k}\) are multiplied by elements contained in proper subfields \(\mathbb{F}_{p^{k/d}}\), and by elements in the base field \(\mathbb{F}_{p}\). We show that significant speedups in pairing computations can be achieved by delaying these “mismatched” multiplications for an optimal number of iterations. Importantly, we show that our technique can be easily integrated into traditional pairing algorithms; implementers can exploit the computational savings herein by applying only minor changes to existing pairing code.


Pairings Miller’s algorithm finite field arithmetic Tate pairing ate pairing 


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  1. 1.
    Arene, C., Lange, T., Naehrig, M., Ritzenthaler, C.: Faster pairing computation. Cryptology ePrint Archive, Report 2009/155 (2009),
  2. 2.
    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
  3. 3.
    Barreto, P.S.L.M., Lynn, B., Scott, M.: Efficient implementation of pairing-based cryptosystems. J. Cryptology 17(4), 321–334 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Barreto, P.S.L.M., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Preneel, B., Tavares, S.E. (eds.) SAC 2005. LNCS, vol. 3897, pp. 319–331. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Benger, N., Scott, M.: Constructing tower extensions for the implementation of pairing-based cryptography. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 180–195. Springer, Heidelberg (2010)Google Scholar
  6. 6.
    Bernstein, D.J.: Curve25519: New diffie-hellman speed records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 207–228. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Bernstein, D.J., Lange, T.: Explicit-formulas database,
  8. 8.
    Boneh, D., Franklin, M.K.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Chatterjee, S., Sarkar, P., Barua, R.: Efficient computation of Tate pairing in projective coordinate over general characteristic fields. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 168–181. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Costello, C., Boyd, C., Nieto, J.M.G., Wong, K.K.-H.: Avoiding full extension field arithmetic in pairing computations. In: Bernstein, D.J., Lange, T. (eds.) AFRICACRYPT 2010. LNCS, vol. 6055, pp. 203–224. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Costello, C., Hisil, H., Boyd, C., Nieto, J.M.G., Wong, K.K.-H.: Faster pairings on special weierstrass curves. In: Shacham, H. (ed.) Pairing 2009. LNCS, vol. 5671, pp. 89–101. Springer, Heidelberg (2009)Google Scholar
  12. 12.
    Costello, C., Lange, T., Naehrig, M.: Faster pairing computations on curves with high-degree twists. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 209–223. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Prem Laxman Das, M., Sarkar, P.: Pairing computation on twisted Edwards form elliptic curves. In: Galbraith, Paterson (eds.) [17], pp. 192–210 (2008)Google Scholar
  14. 14.
    El Mrabet, N., Negre, C.: Finite field multiplication combining AMNS and DFT approach for pairing cryptography. In: Boyd, C., González Nieto, J. (eds.) ACISP 2009. LNCS, vol. 5594, pp. 422–436. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Freeman, D.: Constructing pairing-friendly elliptic curves with embedding degree 10. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 452–465. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. J. Cryptology 23(2), 224–280 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Galbraith, S.D., Paterson, K.G. (eds.): Pairing 2008. LNCS, vol. 5209. Springer, Heidelberg (2008)zbMATHGoogle Scholar
  18. 18.
    Granger, R., Page, D., Stam, M.: On small characteristic algebraic tori in pairing-based cryptography. LMS J. Comput. Math. 9, 64–85 (2006)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Hess, F.: Pairing lattices. In: Galbraith, Paterson (eds.) [17], pp. 18–38Google Scholar
  20. 20.
    Hess, F., Smart, N.P., Vercauteren, F.: The eta pairing revisited. IEEE Transactions on Information Theory 52(10), 4595–4602 (2006)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Hisil, H.: Elliptic Curves, Group Law, and Efficient Computation. PhD thesis, Queensland University of Technology (2010)Google Scholar
  22. 22.
    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), CrossRefGoogle Scholar
  23. 23.
    Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  24. 24.
    Lee, E., Lee, H.-S., Park, C.-M.: Efficient and generalized pairing computation on abelian varieties. IEEE Transactions on Information Theory 55(4), 1793–1803 (2009)CrossRefGoogle Scholar
  25. 25.
    Lin, X., Zhao, C., Zhang, F., Wang, Y.: Computing the ate pairing on elliptic curves with embedding degree k = 9. IEICE Transactions 91-A(9), 2387–2393 (2008)Google Scholar
  26. 26.
    Matsuda, S., Kanayama, N., Hess, F., Okamoto, E.: Optimised versions of the ate and twisted ate pairings. In: Galbraith, S.D. (ed.) Cryptography and Coding 2007. LNCS, vol. 4887, pp. 302–312. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  27. 27.
    Miller, V.S.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17, 235–261 (2004)zbMATHCrossRefGoogle Scholar
  28. 28.
    Montgomery, P.L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Trans. Computers 54(3), 362–369 (2005)zbMATHCrossRefGoogle Scholar
  29. 29.
    Scott, M., Barreto, P.S.L.M.: Compressed pairings. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 140–156. Springer, Heidelberg (2004)Google Scholar
  30. 30.
    Shirase, M., Takagi, T., Choi, D., Han, D.G., Kim, H.: Efficient computation of Eta pairing over binary field with Vandermonde matrix. ETRI journal 31(2), 129–139 (2009)CrossRefGoogle Scholar
  31. 31.
    Vercauteren, F.: Optimal pairings. IEEE Transactions on Information Theory 56(1), 455–461 (2010)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Weimerskirch, A., Paar, C.: Generalizations of the Karatsuba algorithm for efficient implementations. Cryptology ePrint Archive, Report 2006/224 (2006),

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Craig Costello
    • 1
  • Colin Boyd
    • 1
  • Juan Manuel Gonzalez Nieto
    • 1
  • Kenneth Koon-Ho Wong
    • 1
  1. 1.Information Security InstituteQueensland University of TechnologyBrisbaneAustralia

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