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

What Is Order of Element

This section describes the order of a given element in a finite Abelian Group, which is defined as the least positive integer n, such that the scalar multiplication of n and P is 0, where 0 is the identity element.

**What Is the Order of an Element in a Finite Abelian Group?**
The order of an element P in a finite Abelian Group
is the least positive integer n, such that the scalar multiplication of n
and P is 0, where 0 is the identity element.

The above definition is based on the following fact:

For any given element P in a finite Abelian Group, a positive integer m exists, such that the scalar multiplication of m and P is 0. Here is the proof:

Given element P in an finite Abelian Group G a positive integer m exists, such that mP = 0. Proof: Since G is finite, we must have 2 distinct integers i and j, such that iP = jP Otherwise, we will have infinite elements in G. If j > i, we found m: m = j-i and m>0: mP = (j-i)P = 0, because iP = jP If i > j, we found m: m = i-j and m>0: mP = (i-j)P = 0, because iP = jP

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

**Order of Element Example 1.1** -
The order of integer 3 is 8, because:

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

**Order of Element Example 1.2** -
The order of integer 4 is 6, because:

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 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.

**Order of Element Example 2.1** -
The order of Polynomial 1 is 3:

P = 1 2P = 2 3P = 0

**Order of Element Example 2.2** -
The order of Polynomial x is 3:

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

**Order of Element Example 2.3** -
The order of Polynomial x+1 is 3:

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)