# Elliptic Curves in Cryptography

### Fall 2011

## Textbook

## Course Topics

- Introduction to elliptic curves (Washington Chapters 1-2).
- Definition of elliptic curves. Group law. Endomorphisms and isomorphisms.
*j*-invariant. Elliptic curves in SAGE.

- Elliptic curves over finite fields (Ch. 3-4).
- Torsion points. Frobenius morphism. Group structure and group order. Hasse's theorem. Supersingular curves. Discrete logarithm problem.

- Elliptic curve cryptosystems (Ch. 6).
- Diffie-Hellman key exchange. ElGamal encryption. Schnorr identification. ECDSA. Security definitions and proofs.

- Discrete logarithm attacks (Ch. 5).
- Baby step-giant step. Pollard rho. Menezes-Okamoto-Vanstone and Frey-Rück reductions. Anomalous curves.

- Pairing-based cryptography.
- Abstract properties of pairings. Pairing-friendly elliptic curves. Boneh-Franklin identity-based encryption. Boneh-Lynn-Shacham signatures. Boneh-Goh-Nissim homomorphic encryption.

- Generating elliptic curves for cryptography (Ch. 4,10).
- Schoof's algorithm. Complex multiplication. Generating pairing-friendly curves.

- Advanced topics, to be chosen from the following:
- Mathematics of pairings (Ch. 11).
- Elliptic curve factoring and primality testing (Ch. 7).
- Hyperelliptic curves (Ch. 13).
- Weil restriction and index calculus.
- More pairing-based cryptosystems.
- Fast implementations.