*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 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.

High rank elliptic curves with prescribed torsion

Infinite families of elliptic curves with high rank and prescribed torsion

- I. Blake, G. Seroussi, N. Smart:
*Elliptic Curves in Cryptography*, Cambridge University Press, Cambridge, 1999. - J. E. Cremona:
*Algorithms for Modular Elliptic Curves*, Cambridge University Press, Cambridge, 1997. - A. W. Knapp:
*Elliptic Curves*, Princeton University Press, Princeton, 1992. - N. Koblitz:
*A Course in Number Theory and Cryptography*, Springer-Verlag, Berlin, 1994. - N. Koblitz:
*Algebraic Aspects of Cryptography*, Springer-Verlag, Berlin, 1999. - N. Koblitz, A. Menezes, S. A. Vanstone:
*The State of Elliptic Curve Cryptography*, Designs, Codes and Cryptography 19 (2000), pp. 173-193. - J. H. Silverman, J. Tate:
*Rational Points on Elliptic Curves*, Springer-Verlag, Berlin, 1992. - D. R. Stinson:
*Cryptography. Theory and Practice*, CRC Press, Boca Raton, 1996.

- John Cremona's home page
- Ian Connell's APECS
- SIMATH home page
- PARI/GP home page
- Joseph Silverman's home page
- Jean-Francois Mestre's publications
- Koh-ichi Nagao's home page
- Garikai Campbell's home page
- Stefane Fermigier's Page on Elliptic Curves
- Algorithmic Number Theory: Tables and Links
- Elliptic Curves and Cryptology (people, books, theses, software)
- Elliptische Kurven (by Franz Lemmermeyer)
- Number Theory Web