Abelian Group on Elliptic Curve

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.

Cryptography, Abelian Group, Introduction

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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:

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 + 0 = P                   The "identity" condition
P + (-P) = 0                The "symmetry" condition

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

 What Is Abelian Group

 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)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB