Skip to main content
SpringerLink
Log in

On the number of distinct elliptic curves in some families

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We give explicit formulas for the number of distinct elliptic curves over a finite field (up to isomorphism over the algebraic closure of the ground field) in several families of curves of cryptographic interest such as Edwards curves and their generalization due to D. J. Bernstein and T. Lange as well as the curves introduced by C. Doche, T. Icart and D. R. Kohel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Avanzi R., Cohen H., Doche C., Frey G., Lange T., Nguyen K., Vercauteren F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton, FL (2005)

    Google Scholar 

  2. Bernstein D.J., Lange T.: Faster addition and doubling on elliptic curves. In: Proc. Asiacrypt’2007. Lect. Notes in Comp. Sci., vol. 4833, pp. 29–50. Springer, Berlin (2007).

  3. Bernstein D.J., Lange T.: Inverted Edwards coordinates. In: Proc. AAECC’2007. Lect. Notes in Comp. Sci., vol. 4851, pp. 20–27. Springer, Berlin (2007).

  4. Bernstein D.J., Lange T.: Analysis and optimization of elliptic-curve single-scalar multiplication. In: Mullen, G.L., Panario, D., Shparlinski, I.E. (eds) Finite Fields and Applications. Contemp. Math., vol. 461, pp. 1–20. Amer. Math. Soc., Providence, RI (2008)

    Google Scholar 

  5. Bernstein D.J., Birkner P., Joye M., Lange T., Peters C.: Twisted Edwards curves. In: Proc. Africacrypt’2008. Lect. Notes in Comp. Sci., vol. 5023, pp. 389–405. Springer, Berlin (2008).

  6. Bernstein D.J., Lange T., Rezaeian Farashahi R.: Binary Edwards curves. In: CHES’2008. Lect. Notes in Comp. Sci., vol. 5154, pp. 244–265. Springer, Berlin (2008).

  7. Castryck W., Hubrechts H.: The distribution of the number of points modulo an integer on elliptic curves over finite fields. Preprint (2009). Available from http://arxiv.org/abs/0902.4332. Accessed 25 Feb 2009

  8. Cohen S.D.: The distribution of polynomials over finite fields. Acta Arith. 17, 255–271 (1970)

    MATH  MathSciNet  Google Scholar 

  9. Doche C., Icart T., Kohel D.R.: Efficient scalar multiplication by isogeny decompositions. In: PKC’2006. Lect. Notes in Comp. Sci., vol. 3958, pp. 191–206. Springer, Berlin (2006).

  10. Edwards H.M.: A normal form for elliptic curves. Bull. Amer. Math. Soc. 44, 393–422 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  12. Silverman J.H.: The Arithmetic of Elliptic Curves. Springer, Berlin (1995)

    Google Scholar 

  13. von zur Gathen J., Shparlinski I.E.: Computing components and projections of curves over finite fields. SIAM J. Comput. 28, 822–840 (1998)

    Article  Google Scholar 

  14. von zur Gathen J., Karpinski M., Shparlinski I.E.: Counting curves and their projections. Comput. Complex. 6, 64–99 (1996)

    Article  Google Scholar 

  15. Washington L.C.: Elliptic Curves: Number Theory and Cryptography. CRC Press, Boca Raton, FL (2008)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Reza Rezaeian Farashahi & Igor E. Shparlinski

  2. Department of Mathematical Sciences, Isfahan University of Technology, P.O. Box 85145, Isfahan, Iran

    Reza Rezaeian Farashahi

Authors
  1. Reza Rezaeian Farashahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor E. Shparlinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor E. Shparlinski.

Additional information

Communicated by A. Enge.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rezaeian Farashahi, R., Shparlinski, I.E. On the number of distinct elliptic curves in some families. Des. Codes Cryptogr. 54, 83–99 (2010). https://doi.org/10.1007/s10623-009-9310-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-009-9310-2

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation