Andrej Dujella:

Algorithmic Aspects of Elliptic Curves

Faculty of Informatics, University of Debrecen, May 2007


Elliptic curves over Q: Group law. Computing Mordell-Weil group. Construction of curves with high rank.

Elliptic curves over finite fields: Implementation of operations. Algorithms for determining the group order.

Elliptic Curve Cryptography: Public key cryptography based on discrete logarithm problem. Menezes-Vanstone and ECDSA cryptosystems. Elliptic curve discrete logarithm problem. Comparing ECC with other types of cryptography.

Other applications of elliptic curves: Lenstra's elliptic curve factoring method. Elliptic curve primality proving algorithm.

Homework exercises


  1. I. Blake, G. Seroussi, N. Smart: Elliptic Curves in Cryptography, Cambridge University Press, Cambridge, 1999.

  2. I. Connell: Elliptic Curve Handbook, McGill University, 1999.

  3. J. E. Cremona: Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1997.

  4. D. Hankerson, A. Menezes, S. Vanstone: Guide to Elliptic Curve Cryptography, Springer-Verlag, New York, 2004.

  5. A. W. Knapp: Elliptic Curves, Princeton University Press, Princeton, 1992.

  6. N. Koblitz: A Course in Number Theory and Cryptography, Springer-Verlag, Berlin, 1994.

  7. N. Koblitz, A. Menezes, S. A. Vanstone: The State of Elliptic Curve Cryptography, Designs, Codes and Cryptography 19 (2000), pp. 173-193.

  8. J. S. Milne: Elliptic Curves, BookSurge Publishers, 2006.

  9. M. Rosing: Implementing Elliptic Curve Cryptography, Manning, Greenwich, 1999.

  10. S. Schmitt, H.G. Zimmer: Elliptic Curves. A Computational Approach, de Gruyter, Berlin, 2003.

  11. J. H. Silverman: Elliptic curves and cryptography, in: P. Garrett, D. Lieman (Eds.): Public-Key Cryptography, American Mathematical Society, Providence, 2005, pp.91-112.

  12. J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Springer-Verlag, Berlin, 1992.

  13. L.C. Washington: Elliptic Curves: Number Theory and Cryptography, CRC Press, Boca Raton, 2003.

Some useful links

Web pages of some courses on elliptic curves

  1. Elliptic Curves (Edray Goins)

  2. Elliptic Curves (Jim Milne)

  3. Elliptic Curves (Miles Reid)

  4. Elliptic Curves (Helena Verrill)

  5. Elliptic Curves and Applications (Andrej Dujella)

  6. Elliptic Curves and Cryptography (Alan Silvester)

  7. Elliptische Kurven (Franz Lemmermeyer)

  8. Introduction to elliptic curves (Ian Kiming)

  9. Number Theory of Cubic Curves (Jaap Top)

  10. Selected Topics in Number Theory (Jerrold Tunnell)

  11. Vorlesung uber Elliptische Kurven und Kryptographie (Wolfgang Ruppert)

Andrej Dujella home page