CS 259C/Math 250: Elliptic Curves in Cryptography

Elliptic Curves in Cryptography

Fall 2011


Course Topics

  1. Introduction to elliptic curves (Washington Chapters 1-2).
    • Definition of elliptic curves. Group law. Endomorphisms and isomorphisms. j-invariant. Elliptic curves in SAGE.
  2. 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.
  3. Elliptic curve cryptosystems (Ch. 6).
    • Diffie-Hellman key exchange. ElGamal encryption. Schnorr identification. ECDSA. Security definitions and proofs.
  4. Discrete logarithm attacks (Ch. 5).
    • Baby step-giant step. Pollard rho. Menezes-Okamoto-Vanstone and Frey-Rück reductions. Anomalous curves.
  5. 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.
  6. Generating elliptic curves for cryptography (Ch. 4,10).
    • Schoof's algorithm. Complex multiplication. Generating pairing-friendly curves.
  7. 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.