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

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).

- The element set has 9 binary polynomials: 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2.
- The addition operation is the normal algebra addition with coefficients reduced by modulo 3.
- The identity element is the binary polynomial: 0.

**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 ...

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

What Is Subgroup in Abelian Group

►What Is Subgroup Generator in Abelian Group

Every Element Is Subgroup Generator

Order of Subgroup and Lagrange Theorem

Element Generated Subgroup Is Cyclic

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)