Elliptic Curve Theory and Applications: Elliptic Curves Over Finite Fields

​Introductory course on elliptic curves, free and open to the public. This is a technical course, a light background in math will be useful but not necessary.

​Seminar 3: Elliptic Curves over Finite Fields

1. ​Elliptic Curves over F_p

   • ​compressed setup

   • ​finite points

   • ​modular group law

2. ​Finite Group Structure and Torsion

   • ​orders

   • ​cyclic subgroups

   • ​structure of E(F_p)

3. ​Galois, Frobenius, and Point Counting

   • ​field extensions

   • ​Frobenius

   • ​fixed points

   • ​Hasse's Theorem

4. ​From Finite Groups to Cryptography

   • ​scalar multiplication

   • ​ECDLP

   • ​subgroup choice

​Readings:

1. ​Joseph H. Silverman & John T. Tate, "Rational Points on Elliptic Curves" textbook chapters 1-3