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Cryptographic Pairings Based on Elliptic Nets

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Book cover Advances in Information and Computer Security (IWSEC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 7038))

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Abstract

In 2007, Stange proposed a novel method for computing the Tate pairing on an elliptic curve over a finite field. This method is based on elliptic nets, which are maps from ℤn to a ring and satisfy a certain recurrence relation. In the present paper, we explicitly give formulae based on elliptic nets for computing the following variants of the Tate pairing: the Ate, Ate i , R-Ate, and optimal pairings. We also discuss their efficiency by using some experimental results.

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References

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

    Article  MATH  Google Scholar 

  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. 354–369. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Boneh, D., Franklin, M.: Identity-based encryption from the weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–369. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  4. Barreto, P.S.L.M., Lynn, B., Scott, M.: Constructing elliptic curves with prescribed embedding degrees. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 257–267. Springer, Heidelberg (2003)

    Chapter  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)

    Chapter  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. Galbraith, S.D.: Pairings. In: Blake, I., Seroussi, G., Smart, N. (eds.) Advances in Elliptic Curve Cryptography, Ch. IX, Cambridge University Press, Cambridge (2005)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Joux, A.: A one round protocol for tripartite Diffie–Hellman. In: Bosma, W. (ed.) ANTS 2000 Part IV. LNCS, vol. 1838, pp. 385–393. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  10. 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)

    Article  Google Scholar 

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

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

    Article  MATH  Google Scholar 

  13. Murphy, A., Fitzpatrick, N.: Elliptic Curves for Pairing Applications. Cryptology ePrint Archive, Report 2005/302 (2005), http://eprint.iacr.org/2005/302.pdf

  14. Ogura, N., Uchiyama, S., Kanayama, N., Okamoto, E.: A note on the pairing computation using normalized Miller functions, to appear in IEICE Transaction on Fundamentals of Electronics, Communications and Computer Sciences

    Google Scholar 

  15. Sakai, R., Ohgshi, K., Kasahara, M.: Cryptosystems based on pairings. In: Symposium on Cryptography and Information Security 2000, SCIS (2000)

    Google Scholar 

  16. Schoof, R.: Elliptic curves over finite fields and computation of square roots mod p. Math. Comp. 44, 483–494 (1985)

    MATH  Google Scholar 

  17. Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986)

    MATH  Google Scholar 

  18. Stange, K.E.: The tate pairing via elliptic nets. In: Takagi, T., Okamoto, T., Okamoto, E., Okamoto, T. (eds.) Pairing 2007. LNCS, vol. 4575, pp. 329–348. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  19. Taylor, G.: Stange’s algorithm for elliptic nets, http://maths.straylight.co.uk/archives/102

  20. Zhao, C.-A., Zhang, F., Huang, J.: A note on the Ate pairing. International Journal of Information Security 6(7), 379–382 (2008)

    Article  Google Scholar 

  21. Vercauteren, F.: Optimal pairings. IEEE Transactions on Information Theory 56(1), 455–461 (2010)

    Article  Google Scholar 

  22. Ward, M.: Memoir on elliptic divisibility sequence. American Journal of Mathematics 70, 31–74 (1948)

    Article  MATH  Google Scholar 

  23. MAGMA group, Magma, http://magma.maths.usyd.edu.au/magma/

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Naoki Ogura & Shigenori Uchiyama

  2. Faculty of Systems and Information Engineering, University of Tsukuba, 1-1-1, Ten-nohdai, Tsukuba-shi, Ibaraki-ken, 305-8573, Japan

    Naoki Kanayama & Eiji Okamoto

Authors
  1. Naoki Ogura
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  2. Naoki Kanayama
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  3. Shigenori Uchiyama
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  4. Eiji Okamoto
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Editor information

Editors and Affiliations

  1. Dept. of Computational Science and Engineering, Nagoya University, Furo-cho, 464-8603, Chikusa-ku, Nagoya, Japan

    Tetsu Iwata

  2. Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, 432-8011, Naka, Hamamatsu, Japan

    Masakatsu Nishigaki

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Ogura, N., Kanayama, N., Uchiyama, S., Okamoto, E. (2011). Cryptographic Pairings Based on Elliptic Nets. In: Iwata, T., Nishigaki, M. (eds) Advances in Information and Computer Security. IWSEC 2011. Lecture Notes in Computer Science, vol 7038. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25141-2_5

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  • DOI: https://doi.org/10.1007/978-3-642-25141-2_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25140-5

  • Online ISBN: 978-3-642-25141-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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