**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.02, by Dr. Herong Yang

Abelian Group on Elliptic Curve

This section demonstrates that an Abelian Group can be defined with all points on an elliptic curve with the 'rule of chord' operation.

The addition operation we have defined earlier on an elliptic curve actually helps to create an Abelian Group on the curve:

- The set of elements is the set of all points on the curve and the infinity point.
- The binary operation is the addition operation we have defined earlier.
- The identity element is the infinity point.

You can verify that all 5 Abelian Group conditions are satisfied. For example:

P + Q = R The "closure" condition P + Q = Q + P The "commutativity" condition (P + Q) + S = P + (Q + S) The "associativity" condition P + ∞ = P The "identity" condition P + (-P) = ∞ The "symmetry" condition

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

►Abelian Group and Elliptic Curves

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

Modular Multiplication of 11 - Abelian Group

►Abelian Group on Elliptic Curve

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)