Lecture Notes | Elliptic Curves | Mathematics

Some of the theorems and algorithms presented in lecture are demonstrated using Sage, an open-source computer algebra system with extensive support for computing with elliptic curves. You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the Collaborative Calculation in the Cloud.

Please download the PDFs to view them with a PDF reader because some links lead to a non-OCW site that does not allow easy navigation back to the OCW site.

Lecture note files.
LEC #	TOPICS	ASSOCIATED WORKSHEETS
1	Introduction to Elliptic Curves (PDF - 1MB)
2	The Group Law, Weierstrass, and Edwards Equations (PDF)	18.783 Lecture 2: Proof of Associativity (SAGEWS) 18.783 Lecture 2: Group Law on Edwards Curves (SAGEWS)
3	Finite Fields and Integer Arithmetic (PDF)
4	Finite Field Arithmetic (PDF)
5	Isogenies (PDF)
6	Isogeny Kernels and Division Polynomials (PDF)	18.783 Lecture 6: Division Polynomials (SAGEWS)
7	Endomorphism Rings (PDF)
8	Hasse's Theorem, Point Counting (PDF)
9	Schoof's Algorithm (PDF)	18.783 Lecture 9: Schoof's Algorithm (SAGEWS)
10	Generic Algorithms for Discrete Logarithms (PDF)
11	Index Calculus, Smooth Numbers, and Factoring Integers (PDF)	18.783 Lecture 11: Index Calculus (SAGEWS) 18.783 Lecture 11: Pollard P-1 (SAGEWS) 18.783 Lecture 11: Montgomery ECM (SAGEWS)
12	Elliptic Curve Primality Proving (ECPP) (PDF)
13	Endomorphism Algebras (PDF)
14	Ordinary and Supersingular Curves (PDF)
15	Elliptic Curves over C (Part 1) (PDF)
16	Elliptic Curves over C (Part 2) (PDF)
17	Complex Multiplication (PDF)
18	The CM Torsor (PDF)
19	Riemann Surfaces and Modular Curves (PDF)
20	The Modular Equation (PDF)
21	The Hilbert Class Polynomial (PDF)
22	Ring Class Fields and the CM Method (PDF)
23	Isogeny Volcanoes (PDF)
24	Divisors and the Weil Pairing (PDF)
25	Modular Forms and L-Functions (PDF)
26	Fermat's Last Theorem (PDF)