What Is Abelian Group

This section describes Abelian Group, which a set of elements with a binary operation satisfing 5 conditions.

What Is Abelian Group? An Abelian Group is a set, G, of elements with a binary operation, F(), that satisfy the following 5 conditions:

1. Closure - The result, R, of F() of any given two elements, P and Q, in G must be an element in G. In other words:

R = F(P, Q) in G, if P and Q are in G. 

2. Commutativity - The result of the operation is not changing, when the order of given two elements is changed. In other words:

F(P, Q) = F(Q, P) for any given two points.

3. Associativity -

F(F(P, Q), S) = F(P, F(Q, S))

4. Identity Element - A special element called identity element, I, exists with the following property:

F(P, I) = P

5. Symmetry - Every element P has an symmetric element (or inverse element) Q with the following property:

F(P, Q) = I

Good introductions of groups can be found in "Introduction to Finite Fields" by Yunghsiang S. Han at http://web.ntpu.edu.tw/~yshan/algebra.pdf, and in "BSI TR-03111 Elliptic Curve Cryptography, Version 2.10" at bsi.bund.de/SharedDocs/Downloads/EN/BSI/Publications/TechGuidelines/TR03111/BSI-TR-03111_pdf.pdf.

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