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Another Approach to Pairing Computation in Edwards Coordinates

  • Sorina Ionica
  • Antoine Joux
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5365)

Abstract

The recent introduction of Edwards curves has significantly reduced the cost of addition on elliptic curves. This paper presents new explicit formulae for pairing implementation in Edwards coordinates. We prove our method gives performances similar to those of Miller’s algorithm in Jacobian coordinates and is thus of cryptographic interest when one chooses Edwards curve implementations of protocols in elliptic curve cryptography. The method is faster than the recent proposal of Das and Sarkar for computing pairings on supersingular curves using Edwards coordinates.

Keywords

Tate pairing Miller’s algorithm Edwards coordinates 

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References

  1. 1.
    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Lange, T.: Explicit-formulas database (2007), http://hyperelliptic.org/EFD/
  3. 3.
    Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Boneh, D., Lynn, B., Shacham, H.: Short signatures from the Weil pairing. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 514–532. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Boneh, D., Shacham, H.: Group signatures with verifier-local revocation. In: Atluri, V., Pfitzmann, B., McDaniel, P. (eds.) ACM CCS 2004: 11th Conference on Computer and Communications Security, pp. 168–177. ACM Press, New York (2004)Google Scholar
  8. 8.
    Bosma, W.: Signed bits and fast exponentiation. J. de théorie des nombres de Bordeaux 13(1), 27–41 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brezing, F., Weng, A.: Elliptic curves suitable for pairing based cryptography. Des. Codes Cryptography 37(1), 133–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chatterjee, S., Sarkar, P., Barua, R.: Efficient computation of Tate pairing in projective coordinate over general characteristic fields (2004)Google Scholar
  11. 11.
    Das, M.P.L., Sarkar, P.: Pairing computation on twisted Edwards form elliptic curves. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209. Springer, Heidelberg (2008)Google Scholar
  12. 12.
    Edwards, H.M.: A normal form for elliptic curves. Bull. AMS 44, 393–422 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. Cryptology ePrint Archive, Report 2006/372 (2006), http://eprint.iacr.org/
  14. 14.
    Granger, R., Hess, F., Oyono, R., Thériault, N., Vercauteren, F.: Ate pairing on hyperelliptic curves (2007)Google Scholar
  15. 15.
    Granger, R., Page, D., Smart, N.P.: High security pairing-based cryptography revisited. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 480–494. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. 16.
    Hartshorne, R.: Algebraic Geometry. Graduate texts in Mathematics, vol. 52. Springer, Heidelberg (1977)zbMATHGoogle Scholar
  17. 17.
    Hess, F., Smart, N.P., Vercauteren, F.: The Eta Pairing Revisited. IEEE Transactions on Information Theory 52, 4595–4602 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ionica, S., Joux, A.: Another approach on pairing computation in Edwards coordinates. Cryptology ePrint Archive, Report 2008/292 (2008), http://eprint.iacr.org/
  19. 19.
    Joux, A.: A one round protocol for tripartite Diffie-Hellman. Journal of Cryptology 17(4), 263–276 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    Miller, V.S.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate texts in Mathematics, vol. 106. Springer, Heidelberg (1986)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sorina Ionica
    • 2
  • Antoine Joux
    • 1
    • 2
  1. 1.DGAUSA
  2. 2.Université de Versailles Saint-Quentin-en-YvelinesVersailles CEDEXFrance

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