Lecture Notes

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 SageMathCloud™.

1 Introduction to Elliptic Curves (PDF) 18.783 Lecture 1 (SWS)
2 The Group Law, Weierstrass and Edwards Equations (PDF) 18.783 Lecture 2: Proof of Associativity (SWS)
18.783 Lecture 2: Group Law on Edwards Curves (SWS)
3 Integer Arithmetic (PDF)  
4 Finite Field Arithmetic (PDF)  
5 Isogenies and Endomorphisms (PDF)  
6 Division Polynomials and Torsion Subgroups (PDF) 18.783 Lecture 6: Division Polynomials (SWS)
7 Endomorphism Rings and Hasse's Theorem (PDF)  
8 Point Counting (PDF)  
9 Schoof's Algorithm (PDF) 18.783 Lecture 9: Schoof's Algorithm (SWS)
10 Discrete Logarithms: Generic Algorithms (PDF)  
11 Discrete Logarithms: Lower Bounds, Index Calculus (PDF) 18.783 Lecture 11: Index Calculus (SWS)
12 Elliptic Curve Factorization Method (ECM) (PDF) 18.783 Lecture 12: Pollard p-1 (SWS)
13 Elliptic Curve Primality Proving (ECPP) (PDF) 18.783 Lecture 13 Montgomery ECM (SWS)
14 Endomorphism Algebras (PDF)  
15 Ordinary and Supersingular Curves, The j-invariant (PDF)  
16 Elliptic Functions, Eisenstein Series, Weierstrass p-function (PDF)  
17 Complex Tori, Elliptic Curves over C, Lattice j-invariants (PDF)  
18 Uniformization Theorem, Complex Multiplication (PDF)  
19 Orders, Ideals, Class Groups, Isogenies over C (PDF)  
20 Riemann Surfaces and the Modular Curve X(1) (PDF)  
21 Modular Functions and the Modular Equation (PDF)  
22 The Main Theorem of Complex Multiplication (PDF)  
23 CM Method and Isogeny Volcanoes (PDF)  
24 Modular Forms and L-functions (PDF)  
25 Fermat's Last Theorem (PDF) 18.783 Lecture 25 (SWS)