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Created by:

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Last updated:

December 7, 2022

Duration:

Unlimited Duration

FREE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

This course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography.

While this is an introductory course, we will (gently) work our way up to some fairly advanced material, including an overview of the proof of Fermat’s last theorem.

Course Curriculum

    • Introduction to Elliptic Curves (notes) Unlimited
    • Introduction to Elliptic Curves (slides) Unlimited
    • The Group Law and Weierstrass and Edwards Equations (notes) Unlimited
    • Finite Field Arithmetic (notes) Unlimited
    • Finite Field Arithmetic (slides) Unlimited
    • Isogenies (notes) Unlimited
    • Isogenies (slides) Unlimited
    • Isogeny Kernels and Division Polynomials (notes) Unlimited
    • Isogeny Kernels and Division Polynomials (slides) Unlimited
    • Endomorphism Rings (notes) Unlimited
    • Endomorphism Rings (slides) Unlimited
    • Hasse’s Theorem and Point Counting (notes) Unlimited
    • Hasse’s Theorem and Point Counting (slides) Unlimited
    • Schoof’s Algorithm (notes) Unlimited
    • Generic Algorithms for the Discrete Logarithm Problem (notes) Unlimited
    • Index Calculus, Smooth Numbers, and Factoring Integers (notes) Unlimited
    • Index Calculus, Smooth Numbers, and Factoring Integers (slides) Unlimited
    • Elliptic Curve Primality Proving (ECPP) (notes) Unlimited
    • Endomorphism Algebras (notes) Unlimited
    • Endomorphism Algebras (slides) Unlimited
    • Ordinary and Supersingular Curves (notes) Unlimited
    • Elliptic Curves over C (Part I) (notes) Unlimited
    • Elliptic Curves over C (Part II) (notes) Unlimited
    • Complex Multiplication (CM) (notes) Unlimited
    • The CM Torsor (notes) Unlimited
    • Riemann Surfaces and Modular Curves (notes) Unlimited
    • The Modular Equation (notes) Unlimited
    • The Hilbert Class Polynomial (notes) Unlimited
    • Ring Class Fields and the CM Method (notes) Unlimited
    • Isogeny Volcanoes (notes) Unlimited
    • The Weil Pairing (notes) Unlimited
    • Modular Forms and L-Functions (notes) Unlimited
    • Modular Forms and L-Functions (slides) Unlimited
    • Fermat’s Last Theorem (notes) Unlimited

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Massachusetts Institute of Technology
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