Finite Field (Galois Field)

table of content:

• Introduction
• The elements
• ECC

Introduction

In Mathematics, a finite field(Galois Field) is a field that contains a finite number of elements. A Galois field is a set on which Mathematical operations like: Addition, Subtraction, Multiplication and Division are defined and satisfy certain rules. Most common example of a Galois Field is Modular Arithmetic. You can access my previous blog on Modular arithmetic here.

Finite Field is a fundamental topic in the field of Mathematics and Computer Science including Number Theory and Cryptography.

The elements

The number of element of a Galois field is called its order or size. A Finite Field of size q exist if and only if q is prime power (q =  P^k , where p is prime  and k is a positive integer.) 

The simplest example of a finite field are the field of prime orders: For each Prime Number p, the prime field of order q may be constructed as the integer Modulo p, Z/pZ. Where Z the set of real numbers.

Lets take a few examples:

Z is the set of integers

Z = {....,-2, -1, 0, 1, 2, .......}

lets take the prime 7, 9, 13:

Z/7Z = {0, 1, 2, 3, 4, 5, 6}

Z/9Z = {0, 1, 2, 3, 4, 5, 6, 7, 8}

Z/13Z = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

One thing that we notice about the modulo, its set is always p - 1. So, the general equation is:

Z/nZ = {0, 1, 2, ......, (n-1)}

Elliptical Curve Cryptography(ECC)

ECC like Montgomery Curve uses Modular Arithmetic to generate a prime field(which is a square matrix of size, pxp, where p is the prime). This prime field enable use to generates the public and private keys.

More on Montgomery Curve click here.

I will post more on Montgomery Curve. Specifically, on the field of cryptography.