Andrej Dujella:

Elliptic Curves and Applications

AK Zahlentheorie, Technische Universitat Graz, March 2001

Elliptic Curves in Cryptography

Public Key Cryptography. The idea of public key. RSA cryptosystem. Cryptography based on groups.

Elliptic Curves. Addition law. Elliptic curves over finite fields. Hasse's Theorem. Implementation of elliptic curves.

Elliptic Curve Cryptosystems. Analogues of Diffie-Helman, ElGamal and DSA cryptosystems. Comparing elliptic curve with other types of cryptography. Elliptic curve discrete logarithm problem.

Picking Cryptosystem Parameters. Selecting an underlying finite field and an appropriate elliptic curve. Algorithms for determining the group order.

Other Applications of Elliptic Curves. Lenstra's elliptic curve factoring method. Elliptic curve primality proving algorithm.

Elliptic Curves of High Rank

Elliptic Curves over the Rationals. Mordell's theorem. Mazur's Theorem. Birch and Swinerton-Dyer Conjecture.

Torsion Subgroup. Lutz-Nagell Theorem. Construction of curves with prescribed torsion.

Construction of Elliptic Curves with High Rank. Mestre's polynomial method. Finite field method. Computing the rank.

High-rank Curves with Prescribed Torsion. Results of Kulesz, Campbell and Womack. Application of Diophantine m-tuples.

Homework exercises

High rank elliptic curves with prescribed torsion

Infinite families of elliptic curves with high rank and prescribed torsion


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

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

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

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

  5. N. Koblitz: Algebraic Aspects of Cryptography, Springer-Verlag, Berlin, 1999.

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

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

  8. D. R. Stinson: Cryptography. Theory and Practice, CRC Press, Boca Raton, 1996.

Some useful links

Andrej Dujella home page