# CO 789 - Fall 2011

## Topics in Cryptography (Pairing-Based Cryptography)

Instructor: David Jao. Office hours: 3:30-4:30 Wed. & Thurs.

### Course materials

- Course outline (
*https:../../wiki/images/d/df/Co789-2011-outline.pdf*)

### Assignments

- Assignment 1 (
*https:../../wiki/images/a/a6/Co789-2011-a1.pdf*) (due October 14) - Assignment 2 (
*https:../../wiki/images/0/09/Co789-2011-a2.pdf*) (due November 7) - Assignment 3 (
*https:../../wiki/images/8/89/Co789-2011-a3.pdf*) (due December 5)

### Lectures

- September 12: Lattices, periods, elliptic functions, Weierstrass -function
- September 14: Elliptic curves, differential equation for
- September 16: Classification of elliptic functions
- September 26: Divisors, principal divisors
- September 28: Linear spaces
- September 30: Riemann-Roch theorem
- October 3: Linear equivalence, Abel-Jacobi theorem
- October 5: Proof of Riemann-Roch. Elliptic curves from the algebraic viewpoint.
- October 7: Modular curves
- October 12: Elliptic curve group law. Weil pairing definition.
- October 14: Proof of Weil reciprocity
- October 17: Well-definedness and bilinearity of Weil and Tate pairing.
- October 19: Miller's algorithm
- October 21: Proof of non-degeneracy of the Weil pairing
- October 24: Properties of finite fields. Embedding degee.
- October 26: Properties of elliptic curves over finite fields. Balasubramanian-Koblitz theorem
- October 28: Reduced Tate pairing, denominator elimination, MOV reduction
- October 31: Trace map, distortion maps, and symmetric pairings on
*y*^{2}=*x*^{3}+*a**x*for - November 2: Pairings on supersingular elliptic curves
- November 4: Pairings on ordinary elliptic curves. Barreto-Naehrig curves.
- November 7: Cocks-Pinch curves. Complex multiplication algorithm.
- November 9: Ate pairing and optimal Ate pairing.
- November 11: Proof of bilinearity and non-degeneracy for optimal Ate pairings. Optimal pairings on BN curves.
- November 14: Cryptographic pairings and applications. MOV, Tri-partite Diffie-Hellman, BLS. DLOG, DDH, CDH, GapDH, and BDH assumptions.
- November 16: Boneh-Frankin Identity-based encryption. Boneh-Boyen.
- November 18: Public-key encryption with keyword search.
- November 21: Attribute-based encryption
- November 23: Boneh-Goh-Nissim homomorphic encryption
- November 25: Publicly verifiable secret sharing
- November 28: Parallelized Weil pairing
- December 2: Pairing inversion problem
- December 5: Future directions