Elliptic Curves

Lecture Notes and Worksheets

Lecture 1: Introduction to Elliptic Curves

• Introduction to Elliptic Curves (notes) (PDF)
• Introduction to Elliptic Curves (slides) (PDF)

Lecture 2: The Group Law and Weierstrass and Edwards Equations

• The Group Law and Weierstrass and Edwards Equations (notes) (PDF)
• Proof of Associativity (worksheet) (IPYNB)
• Group Law on Edwards Curves (worksheet) (IPYNB)

Lecture 3: Finite Field Arithmetic

• Finite Field Arithmetic (notes) (PDF)
• Finite Field Arithmetic (slides) (PDF)
• Root-Finding in Finite Fields (worksheet) (IPYNB)
• Root-Finding and Polynomial Factorization over Finite Fields (YouTube)

Lecture 4: Isogenies

• Isogenies (notes) (PDF)
• Isogenies (slides) (PDF)

Lecture 5: Isogeny Kernels and Division Polynomials

• Isogeny Kernels and Division Polynomials (notes) (PDF)
• Isogeny Kernels and Division Polynomials (slides) (PDF)
• Division Polynomials (worksheet) (IPYNB)

Lecture 6: Endomorphism Rings

• Endomorphism Rings (notes) (PDF)
• Endomorphism Rings (slides) (PDF)

Lecture 7: Hasse’s Theorem and Point Counting

• Hasse’s Theorem and Point Counting (notes) (PDF)
• Hasse’s Theorem and Point Counting (slides) (PDF)

Lecture 8: Schoof’s Algorithm

• Schoof’s Algorithm (notes) (PDF)
• Schoof’s Algorithm (worksheet) (IPYNB)

Lecture 9: Generic Algorithms for the Discrete Logarithm Problem

• Generic Algorithms for the Discrete Logarithm Problem (notes) (PDF)

Lecture 10: Index Calculus, Smooth Numbers, and Factoring Integers

• Index Calculus, Smooth Numbers, and Factoring Integers (notes) (PDF)
• Index Calculus, Smooth Numbers, and Factoring Integers (slides) (PDF)
• Index Calculus (worksheet) (IPYNB)
• Montgomery Elliptic Curve Factorization Method (worksheet) (IPYNB)
• Simple Implementation of Pollard p-1 Algorithm (worksheet) (IPYNB)

Lecture 11: Elliptic Curve Primality Proving (ECPP)

• Elliptic Curve Primality Proving (ECPP) (notes) (PDF)

Lecture 12: Endomorphism Algebras

• Endomorphism Algebras (notes) (PDF)
• Endomorphism Algebras (slides) (PDF)

Lecture 13: Ordinary and Supersingular Curves

• Ordinary and Supersingular Curves (notes) (PDF)

Lecture 14: Elliptic Curves over C (Part I)

• Elliptic Curves over C (Part I) (notes) (PDF)

Lecture 15: Elliptic Curves over C (Part II)

• Elliptic Curves over C (Part II) (notes) (PDF)

Lecture 16: Complex Multiplication (CM)

• Complex Multiplication (CM) (notes) (PDF)

Lecture 17: The CM Torsor

• The CM Torsor (notes) (PDF)

Lecture 18: Riemann Surfaces and Modular Curves

• Riemann Surfaces and Modular Curves (notes) (PDF)

Lecture 19: The Modular Equation

• The Modular Equation (notes) (PDF)

Lecture 20: The Hilbert Class Polynomial

• The Hilbert Class Polynomial (notes) (PDF)

Lecture 21: Ring Class Fields and the CM Method

• Ring Class Fields and the CM Method (notes) (PDF)

Lecture 22: Isogeny Volcanoes

• Isogeny Volcanoes (notes) (PDF)

Lecture 23: The Weil Pairing

• The Weil Pairing (notes) (PDF)

Lecture 24: Modular Forms and L-Functions

• Modular Forms and L-Functions (notes) (PDF)
• Modular Forms and L-Functions (slides) (PDF)

Lecture 25: Fermat’s Last Theorem

• Fermat’s Last Theorem (notes) (PDF)