Today I finally submitted my diploma thesis. It is titled *Mathematical Foundations of Elliptic Curve Cryptography* and contains the following topics:

- A short introduction to algebraic curves (including Riemann-Roch, but without proofs);
- some parts of the general theory elliptic curves (including basics about the Tate module and the Weil and Tate pairings);
- elliptic curves over the complex numbers;
- elliptic curves over finite fields (Weil conjectured for elliptic curves);
- elliptic curves over local fields (and the canonical lift);
- division polynomials;
- modular polynomials (also over finite fields);
- an introduction to elliptic curve cryptography;
- point counting (SEA and Satoh’s algorithm);
- cryptoanalysis: MOV/Frey-Rück, anomalous curve and Weil descent attacks
- and some connections to elliptic divisibility sequences.

Now all that is left to do is to prepare for the diploma examination…

**Tags:** cryptography, diploma thesis, ECC, elliptic curve, math.AG, math.NT