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

Order of Subgroup and Lagrange Theorem

This section describes Lagrange Theorem which states that the order of any subgroup in an finite Abelian group divides the order of the parent group.

**What Is Lagrange Theorem?**
Lagrange Theorem states that the order of any subgroup
in an finite Abelian group divides the order of the parent group.

Lagrange Theorem can be expressed as:

Given an finite Abelian group G with an order of l, and subgroup S with an order of n, there is an integer factor x exists, such that: n*x = l

The factor x in the above expression is also called the cofactor of subgroup S.

The proof of Lagrange Theorem is given in "Lagrange's theorem (group theory)" at wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory).

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

**Order of Subgroup Example 1.1** -
The order of the subgroup generated by 3 is 8, which divides 24 as
8*3 = 24:

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

**Order of Subgroup Example 1.2** -
The order of the subgroup generated by 4 is 6, which divides 24 as
6*4 = 24:

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

**Group Example 2** -
Let's look at the multiplicative Abelian group of the Binary Field
GF(3^{2})/(x^{2}+1).
The order of the group is 9.

**Order of Subgroup Example 2.1** -
The order of the subgroup generated by Polynomial 1 is 3,
which divides 9 as 3*3 = 9:

P = 1 2P = 2 3P = 0

**Order of Subgroup Example 2.2** -
The order of the subgroup generated by Polynomial x is 3,
which divides 9 as 3*3 = 9:

P = x 2P = 2x 3P = 0

**Order of Subgroup Example 2.3** -
The order of the subgroup generated by Polynomial x+1 is 3,
which divides 9 as 3*3 = 9:

P = x+1 2P = 2x+2 3P = 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)