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Mathematical Problems in Engineering
Volume 2013, Article ID 136767, 8 pages
http://dx.doi.org/10.1155/2013/136767
Research Article

The Pairing Computation on Edwards Curves

1College of Sciences, North China University of Technology, Beijing 100144, China
2School of Mathematical Sciences, Peking University, Beijing 100871, China
3Beijing International Center for Mathematical Research, Beijing 100871, China

Received 4 May 2013; Accepted 22 September 2013

Academic Editor: Jun Jiang

Copyright © 2013 Hongfeng Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Avanzi, H. Cohen, C. Doche et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography, CRC Press, 2005. View at MathSciNet
  2. I. F. Blake, G. Seroussi, and N. P. Smart, Advances in Elliptic Curve Cryptography, Cambridge University Press, Cambridge, UK, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  3. V. S. Miller, “The weil pairing, and its efficient calculation,” Journal of Cryptology, vol. 17, no. 4, pp. 235–261, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  4. C. Arène, T. Lange, M. Naehrig, and C. Ritzenthaler, “Faster computation of the Tate pairing,” Journal of Number Theory, vol. 131, no. 5, pp. 842–857, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  5. F. Hess, N. P. Smart, and F. Vercauteren, “The Eta pairing revisited,” IEEE Transactions on Information Theory, vol. 52, no. 10, pp. 4595–4602, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. F. Hess, “Pairing lattices,” in Pairing-Based Cryptography—Pairing 2008, vol. 5209 of Lecture Notes in Computer Science, pp. 18–38, Springer, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. N. Koblitz and A. Menezes, “Pairing-based cryptography at high security levels,” in Cryptography and Coding, vol. 3796 of Lecture Notes in Computer Science, pp. 13–36, Springer, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. F. Vercauteren, “Optimal pairings,” IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 455–461, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. H. M. Edwards, “A normal form for elliptic curves,” Bulletin of the American Mathematical Society, vol. 44, no. 3, pp. 393–422, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. D. J. Bernstein and T. Lange, “Faster addition and doubling on elliptic curves,” in Advances in Cryptology—ASIACRYPT 2007, K. Kurosawa, Ed., vol. 4833 of Lecture Notes in Computer Science, pp. 29–50, Springer, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. D. J. Bernstein, P. Birkner, M. Joye, T. Lange, and C. Peters, “Twisted Edwards curves,” in Progress in Cryptology—AFRICACRYPT 2008, vol. 5023 of Lecture Notes in Computer Science, pp. 389–405, Springer, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. P. L. Das and P. Sarkar, “Pairing computation on twisted Edwards form elliptic curves,” in Pairing-Based Cryptography—Pairing 2008, vol. 5209 of Lecture Notes in Computer Science, pp. 192–210, Springer, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Ionica and A. Joux, “Another approach to pairing computation in Edwards coordinates,” in Progress in Cryptology—INDOCRYPT 2008, vol. 5365 of Lecture Notes in Computer Science, pp. 400–413, Springer, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Yasuda, T. Takagi, and K. Sakurai, “Application of Scalar multiplication of Edwards curves to pairing-based cryptography,” in Advances in Information and Computer Security, vol. 7631 of Lecture Notes in Computer Science, pp. 19–36, Springer, 2012. View at Google Scholar
  15. J. R. Merriman, S. Siksek, and N. P. Smart, “Explicit 4-descents on an elliptic curve,” Acta Arithmetica, vol. 77, no. 4, pp. 385–404, 1996. View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. Costello, T. Lange, and M. Naehrig, “Faster pairing computations on curves with high-degree twists,” in Public Key Cryptography—PKC 2010, vol. 6056 of Lecture Notes in Computer Science, pp. 224–242, Springer, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. D. Galbraith, Mathematics of Public Key Cryptography, Cambridge University Press, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. H. Hisil, K. K.-H. Wong, G. Carter, and E. Dawson, “Twisted Edwards curves revisited,” in Advances in Cryptology—ASIACRYPT 2008, vol. 5350 of Lecture Notes in Computer Science, pp. 326–343, Springer, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. D. J. Bernstein and T. Lange, “A complete set of addition laws for incomplete Edwards curves,” Journal of Number Theory, vol. 131, no. 5, pp. 858–872, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. S. D. Galbraith, X. Lin, and M. Scott, “Endomorphisms for faster elliptic curve cryptography on a large class of curves,” Journal of Cryptology, vol. 24, no. 3, pp. 446–469, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus