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A note on the Ate pairing

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Abstract

The Ate pairing has been suggested since it can be computed efficiently on ordinary elliptic curves with small values of the traces of Frobenius t. However, not all pairing-friendly elliptic curves have this property. In this paper, we generalize the Ate pairing and find a series of the variations of the Ate pairing. We show that the shortest Miller loop of the variations of the Ate pairing can possibly be as small as r 1/φ(k) on some special pairing-friendly curves with large values of Frobenius trace, and hence speed up the pairing computation significantly.

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References

  1. Barreto, P.S.L.M., Galbraith, S., ÓhÉigeartaigh, C., Scott, M.: Efficient pairing computation on supersingular Abelian varieties. In: Designs, Codes and Cryptography, vol. 42, no. 3, pp. 239–271. Springer Netherlands (2007)

  2. Barreto, P.S.L.M., Kim, H.Y., Lynn, B., Scott, M.: Efficient algorithms for pairing-based cryptosystems. In: Advances in Cryptology-Crypto’2002. Lecture Notes in Computer Science vol. 2442, pp. 354–368. Springer Heidelberg (2002)

  3. Barreto, P.S.L.M., Naehrig, M.: Pairing-friendly elliptic curves of prime order. In: Proceedings of SAC 2005-Workshop on Selected Areas in Cryptography. Lecture Notes in Computer Science vol. 3897, pp. 319–331. Springer Heidelberg (2006)

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Duan, P., Cui, S., Chan, C.W.: Special polynomial families for generating more suitable elliptic curves for pairing-based cryptosystems. In: The 5th WSEAS International Conference on Electronics, Hardware, Wireless and Optimal Communications http://eprint.iacr.org/2005/342 (2006)

  6. Duursma, I., Lee, H.-S.: Tate pairing implementation for hyperelliptic curves y 2 = x p − x + d. In: Advances in Cryptology-Asiacrypt’2003. Lecture Notes in Computer Science vol. 2894, pp. 111–123. Springer Heidelberg (2003)

  7. Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. http://eprint.iacr.org/2006/372 (2006, Preprint)

  8. Frey G., Rück H.-G.: A remark concerning m-divisibility and the discrete logartihm in the divisor class group of curves. Math. Comp. 62(206), 865–874 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Galbraith, S., Harrison, K., Soldera, D.: Implementing the Tate pairing. In: Algorithm Number Theory Symposium ANTS V. Lecture Notes in Computer Science vol. 2369, pp. 324–337. Springer Heidelberg (2002)

  10. Galbraith S.: Pairings—Advances in Elliptic Curve Cryptography. Cambridge University Press, London (2005)

    Google Scholar 

  11. Hess F., Smart N.P., Vercauteren F.: The Eta pairing revisited. IEEE Trans. Inf. Theory 52, 4595–4602 (2006)

    Article  MathSciNet  Google Scholar 

  12. Joux, A.: A one round protocol for tripartite DiffieCHellman. In: ANTS-4: Algorithmic Number Theory. Lecture Notes in Computer Science vol. 1838, pp. 385–394. Springer Heidelberg (2000)

  13. Joux, A.: The Weil and Tate pairings as building blocks for public key cryptosystems. In: ANTS-5: Algorithmic Number Theory. Lecture Notes in Computer Science vol. 2369, pp. 20–32. Springer Heidelberg (2002)

  14. Kachisa, E.J., Schaefer, E.F., Scott, M.: Constructing Brezing-Weng pairing friendly elliptic curves using elements in the cyclotomic field. Preprint, 2007. http://eprint.iacr.org/2007/452 (2007, prepaid)

  15. Lidl, R., Niederreiter, H.: Finite Fields. In: Encyclopedia of Mathematics and its Applications, 2nd edn. no. 20, Cambridge University Press, Cambridge (1997)

  16. Matsuda, S., Kanayama, N., Hess, F., Okamoto, E.: Optimised versions of the Ate and twisted Ate pairings. In: The 11th IMA International Conference on Cryptography and Coding. Lecture Notes in Computer Science vol. 4887, pp. 302–312. Springer Heidelberg http://eprint.iacr.org/2007/013 (2007)

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

  18. Murphy, A., Fitzpatrick, N.: Elliptic curves for pairing applications. http://eprint.iacr.org/2005/302 (2005, Preprint)

  19. Naehrig, M., Barreto, P.S.L.M.: On compressible pairings and their computation. http://eprint.iacr.org/2007/429 (2007, Preprint)

  20. Paterson K.G.: Cryptography from Pairing—Advances in Elliptic Curve Cryptography. Cambridge University Press, London (2005)

    Google Scholar 

  21. Sakai, R., Ohgishi, K., Kasahara, M.: Cryptosystems based on pairing. In: Proceedings of 2000 Symposium on Cryptography and Information Security-SCIS 2000, pp. 26–28, Okinawa, Japan, January 2000

  22. Scott, M.: Implementing cryptographic pairings. In: the 10th Workshop on Elliptic Curve Cryptography (2006)

  23. Silverman, J.H.: The arithmetic of elliptic curves. In: Graduate Texts in Mathematics no. 106. Springer, New York (1986)

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Authors and Affiliations

  1. School of Computer Science and Educational Software, Guangzhou University, 510006, Guangzhou, People’s Republic of China

    Chang-An Zhao

  2. School of Information Science and Technology, SunYat-Sen University, 510275, Guangzhou, People’s Republic of China

    Chang-An Zhao, Fangguo Zhang & Jiwu Huang

  3. Guangdong Key Laboratory of Information Security Technology, SunYat-Sen University, 510275, Guangzhou, People’s Republic of China

    Fangguo Zhang & Jiwu Huang

Authors
  1. Chang-An Zhao
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  2. Fangguo Zhang
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  3. Jiwu Huang
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Correspondence to Chang-An Zhao.

Additional information

This work is supported by the National Natural Science Foundation of China (No. 60773202, 60633030) and 973 Program (No. 2006CB303104).

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Zhao, CA., Zhang, F. & Huang, J. A note on the Ate pairing. Int. J. Inf. Secur. 7, 379–382 (2008). https://doi.org/10.1007/s10207-008-0054-1

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