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