**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by Herong Yang

Abelian Group and Elliptic Curves

This chapter provides an introduction to Abelian Group, which can be expressed in the multiplicative notation or the additive notation. An Abelian Group and be defined on an elliptic curve using the 'rule of chord' operation.

Niels Henrik Abel and Abelian Group

Multiplicative Notation of Abelian Group

Additive Notation of Abelian Group

Modular Addition of 10 - Abelian Group

Modular Multiplication of 10 - Not Abelian Group

Takeaways:

- An Abelian Group is a set of elements and a binary operation that satisfy 5 conditions.
- An Abelian Group can be expressed in the multiplicative notation.
- All rational numbers with the multiplication operation is an Abelian Group, and an good example for the multiplicative notation.
- An Abelian Group can be expressed in the additive notation.
- The modular arithmetic addition of any non-negative integer i can be used to define an Abelian Group, G(i,+).
- The modular arithmetic multiplication of any prime integer p can also be used to define an Abelian Group, G(p,*).
- An Abelian Group can be defined with all points on an elliptic curve with the "rule of chord" operation.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

►Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)