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).

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

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