MAC 6396: Elliptic Curves
The course assumes familiarity with elementary concepts from algebra, as covered by an introductory algebra course, for instance. After completion of this course you should be acquainted with the basic definitions and results from the theory of elliptic curves. You should understand and be able to explain essential algebraic properties of elliptic curves over fields of different characteristics. Moreover, after completion of this course, you should know how to perform basic algorithmic tasks related to elliptic curves, like computing in a group defined by an elliptic curve, evaluating a Weil pairing or counting the number of points on an elliptic curve.
Most of the material will be taken from the book Elliptic Curves: Number Theory and Cryptography (Lawrence C. Washington, Chapman & Hall/CRC, 2003), subsequently referred to as [Was03].
The following topics that are to be discussed:
- Basic theory of elliptic curves
- Torsion points
- Elliptic curves over finite fields
- Elliptic curves over the rational numbers
- Complex multiplication
- Divisors
- Zeta functions
More information on the course is available in the syllabus, and comments are welcome.
Topics Discussed in Class
- 08/27/07: Weierstrass equations
Literature: [Was03, Ch. 2.1]
- 08/29/07: Geometric interpretation of the group law
Literature: [Was03, Ch. 2.2]
- 08/31/07: projective plane and intersection multiplicity
Literature: [Was03, Ch. 2.3, Ch. 2.4]
- 09/05/07: singular points, j-invariant
Literature: [Was03, Ch. 2.4, Ch. 2.6]
- 09/07/07: j-invariant, endomorphisms
Literature: [Was03, Ch. 2.6, Ch. 2.8]
- 09/10/07: endomorphisms - examples; Quiz 1
Litrature: [Was03, Ch. 2.8]
- 09/12/07: kernel size of a separable endomorphism
Literature: [Was03, Ch. 2.8]
- 09/14/07: surjectivity of endomorphisms over an algebraically closed field
Literature: [Was03, Ch. 2.8]
- 09/17/07: basic idea of defining projective planes over non-fields; torsion points; Homework 1
Literature: [Was03, Ch. 2.10, Ch. 3.1]
- 09/19/07: division polynomials
Literature: [Was03, Ch. 3.2]
- 09/21/07: structure of E[n]
Literature: [Was03, Ch. 3.2]
- 09/24/07: properties of the Weil pairing; Joux's tripartite Diffie-Hellman protocol
Literature: [Was03, Ch. 3.3]; A. Joux: A One Round Protocol for Tripartite Diffie-Hellman
- 09/26/07: properties of the Weil pairing; structure of E(GF(q))
Literature: [Was03, Ch. 3.3, Ch. 4.1]
- 09/28/07: Hasse's theorem
Literature: [Was03, Ch. 4.2]
- 10/01/07: properties of the Frobenius endomorphism; Quiz 2
Literature: [Was03, Ch. 4.2]
- 10/03/07: computing the group order of an elliptic curve
Literature: [Was03, Ch. 4.3]
- 10/05/07: zeta function of an ellipic curve over GF(q)
Literature: [Was03, Ch. 12.1]
- 10/08/07: some properties of supersingular elliptic curves
Literature: [Was03, Ch. 4.6]
- 10/10/07: divisors, uniformizers
Literature: [Was03, Ch. 11.1], [J. H. Silverman: The Arithmetic of Elliptic Curves, Ch. 2.1]
- 10/12/07: principal divisors
Literature: [Was03, Ch. 11.1]
- 10/15/07: principal divisors
Literature: [Was03, Ch. 11.1]
- 10/17/07: definition of the Weil pairing
Literature: [Was03, Ch. 11.2]
- 10/19/07: computing the Weil pairing; Quiz 3
Literature: [Was03, Ch. 11.4]
- 10/22/07: proving properties of the Weil pairing
Literature: [Was03, Ch. 11.2]
- 10/24/07: MOV attack; Homework 2
Literature: [Was03, Ch. 5.3]
- 10/26/07: Applying the Weil pairing to the Decision Diffie Hellman problem
Literature: [Was03, Ch. 6.2]
- 10/29/07: solution of Quiz 3
- 10/31/07: identity based encryption using a pairing on an elliptic curve
Literature: [Was03, Ch. 6.8]
- 11/02/07: an elliptic curve based encryption scheme relying on the hardness of factoring integers
Literature: [Was03, Ch. 6.7]
- 11/05/07: Riemann-Roch Theorem
Literature: [Was03, Ch. 11.5]
- 11/07/07: genus of of a nonsingular algebraic curve
Literature: [Was03, Ch. 11.5]
- 11/09/07: anomalous elliptic curves
Literature: [Was03, Ch. 5.4]
- 11/14/07: anomalous elliptic curves
Literature: [Was03, Ch. 5.4]
- 11/16/07: computing discrete logarithms on anomalous elliptic curves
Literature: [Was03, Ch. 5.4, Ch. 8.1]
- 11/19/07: Lutz-Nagell Theorem
Literature: [Was03, Ch. 8.1]
- 11/21/07: Lutz-Nagell Theorem
Literature: [Was03, Ch. 8.1]
- 11/23/07: complex multiplication
Literature: [Was03, Ch. 10.2]
- 11/26/07: short introduction to p-adic numbers
Literature: [Was03, Appendix A]
- 11/28/07: review for the final exam
- 11/30/07: final exam
- 12/03/07: endomorphism ring of an elliptic curve over GF(p)
Literature: [Was03, Ch. 10.2]
For questions or comments, please feel free to contact me anytime
(see my homepage for email, phone number, etc.).
Dec 4, 2007