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

What Is Subgroup in Abelian Group

This section describes Subgroups in a Abelian Group. A subgroup in a Abelian Group is a subset of the Abelian Group that itself is an Abelian Group. The subgroup and its parent group are using the same operation.

**What Is Subgroup in Abelian Group?**
A subgroup in an Abelian Group is a subset of the Abelian Group that
itself is an Abelian Group. The subgroup and its parent group are
using the same operation.

**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 Example 1.1** -
Here is a subgroup in the above group:

- The element set has 8 integers: 0, 3, 6, 9, 12, 15, 18, 21.
- The operation is still the same addition operation with modular reduction of 24.
- The identity element is the same: 0.

**Subgroup Example 1.2** -
Here is another subgroup in the above group:

- The element set has 6 integers: 0, 4, 8, 12, 16, 20.
- The operation is still the same addition operation with modular reduction of 24.
- The identity element is the same: 0.

**Group Example 2** -
Here is 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 Example 2.1** -
Here is a subgroup in the above group:

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

The table below defines the addition operation of this subgroup:

+ | 0 1 2 ----------------------- 0 | 0 1 2 1 | 1 2 0 2 | 2 0 1

**Subgroup Example 2.2** -
Here is another subgroup in the above Abelian Group:

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

The table below defines the addition operation of this subgroup:

+ | 0 x 2x ----------------------- 0 | 0 x 2x x | x 2x 0 2x | 2x 0 x

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)