Twisted Edwards Curves

  • Daniel J. Bernstein
  • Peter Birkner
  • Marc Joye
  • Tanja Lange
  • Christiane Peters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5023)

Abstract

This paper introduces “twisted Edwards curves,” a generalization of the recently introduced Edwards curves; shows that twisted Edwards curves include more curves over finite fields, and in particular every elliptic curve in Montgomery form; shows how to cover even more curves via isogenies; presents fast explicit formulas for twisted Edwards curves in projective and inverted coordinates; and shows that twisted Edwards curves save time for many curves that were already expressible as Edwards curves.

Keywords

Elliptic curves Edwards curves twisted Edwards curves Montgomery curves isogenies 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernstein, D.J.: Curve25519: New Diffie-Hellman Speed Records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 207–228. Springer, Heidelberg (2006), http://cr.yp.to/papers.html#curve25519 CrossRefGoogle Scholar
  2. 2.
    Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: ECM using Edwards curves (2007) (Citations in this document: §1), http://eprint.iacr.org/2008/016
  3. 3.
    Bernstein, D.J., Lange, T.: Explicit-formulas database (2007) (Citations in this document: §5, §6), http://hyperelliptic.org/EFD
  4. 4.
    Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Asiacrypt 2007 [19], pp. 29–50 (2007) (Citations in this document: §1, §1, §2, §2, §2, §3, §3, §3, §3, §3, §3, §4, §6, §6, §6, §7, §7), http://cr.yp.to/papers.html#newelliptic
  5. 5.
    Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: AAECC 2007 [8], pp. 20–27 (2007) (Citations in this document: §1, §6, §7), http://cr.yp.to/papers.html#inverted
  6. 6.
    Billet, O., Joye, M.: The Jacobi model of an elliptic curve and side-channel analysis. In: AAECC 2003 [14], pp. 34–42 (2003) MR 2005c:94045 (Citations in this document: §5), http://eprint.iacr.org/2002/125
  7. 7.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. Cambridge University Press, Cambridge (2000) (Citations in this document: §5)Google Scholar
  8. 8.
    Boztaş, S., Lu, H.-F(F.) (eds.): AAECC 2007. LNCS, vol. 4851. Springer, Heidelberg (2007)Google Scholar
  9. 9.
    Brier, É., Joye, M.: Fast point multiplication on elliptic curves through isogenies. In: AAECC 2003 [14], pp. 43–50 (2003) (Citations in this document: §5)Google Scholar
  10. 10.
    Cohen, H., Frey, G. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. CRC Press, Boca Raton (2005) See [11] Google Scholar
  11. 11.
    Doche, C., Lange, T.: Arithmetic of elliptic curves. In: [10] (2005), pp. 267– 302. MR 2162729 (Citations in this document: §3)Google Scholar
  12. 12.
    Duquesne, S.: Improving the arithmetic of elliptic curves in the Jacobi model. Information Processing Letters 104, 101–105 (2007) (Citations in this document: §5)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Edwards, H.M.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44, 393–422 (2007) (Citations in this document: §1), http://www.ams.org/bull/2007-44-03/S0273-0979-07-01153-6/home.html MATHCrossRefGoogle Scholar
  14. 14.
    Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.): AAECC 2003. LNCS, vol. 2643. Springer, Heidelberg (2003) See [6], [9]MATHGoogle Scholar
  15. 15.
    Galbraith, S.D., McKee, J.: The probability that the number of points on an elliptic curve over a finite field is prime. Journal of the London Mathematical Society 62, 671–684 (2000) (Citations in this document: §4), http://www.isg.rhul.ac.uk/~sdg/pubs.html MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: INDOCRYPT 2007 [23] (2007) (Citations in this document: §5) Google Scholar
  17. 17.
    Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster group operations on elliptic curves. 25 Feb 2008 version (2008) (Citations in this document: §5), http://eprint.iacr.org/2007/441
  18. 18.
    Imai, H., Zheng, Y. (eds.): PKC 2000. LNCS, vol. 1751. Springer, Heidelberg (2000) see [21]MATHGoogle Scholar
  19. 19.
    Kurosawa, K. (ed.): ASIACRYPT 2007. LNCS, vol. 4833. Springer, Heidelberg (2007)MATHGoogle Scholar
  20. 20.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987), (Citations in this document: §3, §7), http://links.jstor.org/sici?sici=0025-571819870148:177243:STPAEC2.0.CO;2-3
  21. 21.
    Okeya, K., Kurumatani, H., Sakurai, K.: Elliptic curves with the Montgomery-form and their cryptographic applications. In: PKC 2000, pp. 238–257 (2000) (Citations in this document: §3 §3)Google Scholar
  22. 22.
    Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics 106 (1986)Google Scholar
  23. 23.
    Srinathan, K., Rangan, C.P., Yung, M. (eds.): INDOCRYPT 2007. LNCS, vol. 4859. Springer, Heidelberg (2007) See [16]MATHGoogle Scholar
  24. 24.
    Stein, W. (ed.): Sage Mathematics Software (Version 2.8.12), The Sage Group (2008) (Citations in this document: §3), http://www.sagemath.org
  25. 25.
    Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.): PKC 2006. LNCS, vol. 3958. Springer, Heidelberg (2006) See [1]MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Daniel J. Bernstein
    • 1
  • Peter Birkner
    • 2
  • Marc Joye
    • 3
  • Tanja Lange
    • 2
  • Christiane Peters
    • 2
  1. 1.Department of Mathematics, Statistics, and Computer Science (M/C 249)University of Illinois at ChicagoChicagoUSA
  2. 2.Department of Mathematics and Computer ScienceTechnische Universiteit EindhovenEindhovenNetherlands
  3. 3.Thomson R&D France, Technology Group, Corporate Research, Security LaboratoryCesson-Sévigné CedexFrance

Personalised recommendations