What Is Subgroup Generator in Abelian Group

This section describes subgroup generator in a Abelian Group. A subgroup generator is an element in an Abelian Group that can be used to generator a subgroup using a series of scalar multiplication operations.

What Is Subgroup Generator in Abelian Group? A subgroup generator is an element in an Abelian Group that can be used to generator a subgroup using a series of scalar multiplication operations as defined below in additive notation:

Given an element P in an Abelian Group G, 
if P, 2P, 3P, ..., is a subgroup S, 
P is called the generator of subgroup S.

Group Example 1 - Here is the additive Abelian group of integers: 0, 1, 2, ..., 23 and the addition operation with modular reduction of 24.

Subgroup Generator Example 1.1 - Integer 3 is the generator of the subgroup of 0, 3, 6, 9, 12, 15, 18, 21. because:

 
P = 3
2P = 6
3P = 9
4P = 12
5P = 15
6P = 18
7P = 21
8P = 0
9P = 3
...

Subgroup Generator Example 1.2 - Integer 4 is the generator of the subgroup of 0, 4, 8, 12, 16, 20, because:

 
P = 4
2P = 8
3P = 12
4P = 16
5P = 20
6P = 0
7P = 4
...

Group Example 2 - Let's look at the multiplicative Abelian group of the Binary Field GF(3^2)/(x^2+1).

Subgroup Generator Example 2.1 - Polynomial 1 is a generator of the subgroup of 0, 1, 2, because:

 
P = 1
2P = 2
3P = 0
4P = 1
5P = 2
6P = 0
...

Subgroup Generator Example 2.2 - Polynomial x is a generator of the subgroup of 0, x, 2x, because:

P = x
2P = 2x
3P = 0
4P = x
5P = 2x
6P = 0
...

Subgroup Generator Example 2.3 - Polynomial x+1 is a generator of the subgroup of 0, x+1, 2x+2, because:

P = x+1
2P = 2x+2
3P = 0
4P = x+1
5P = 2x+2
6P = 0
...

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

Generators and Cyclic Subgroups

 What Is Subgroup in Abelian Group

What Is Subgroup Generator in Abelian Group

 What Is Order of Element

 Every Element Is Subgroup Generator

 Order of Subgroup and Lagrange Theorem

 What Is Cyclic Group

 Element Generated Subgroup Is Cyclic

 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