Galois fields,  named after Evariste Galois also known as Finite Field,  is a mathematical concept in abstract algebra that deals with finite mathematical structures. It is a set of numbers that consists of a finite number of elements and has two operations, addition and multiplication, that follow specific rules. The rules for these operations ensure that the Galois Field remains closed, meaning that the result of any operation performed within the set will also be an element of the set.

Galois Fields are useful in various fields, such as cryptography, coding theory, and error correction, due to their unique mathematical properties.

The size of a Galois Field is represented by a prime number ‘p’, and it is denoted by GF(p), where p is a prime number.

Example: One example of a Galois Field is a field with 2 elements, denoted by GF(2). This field has two elements, 0 and 1, and the rules for addition and multiplication operations are defined as follows:

1.   Addition: The addition operation in GF(2) is equivalent to the XOR operation. For example, 0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 0.
2.   Multiplication: The multiplication operation in GF(2) is equivalent to the AND operation. For example, 0 * 0 = 0, 0 * 1 = 0, and  1 * 1 = 1.

Another example of a Galois Field is GF(3), which has 3 elements, 0, 1, and 2. The addition and multiplication operations in this field are performed modulo 3, meaning that the result of any operation will always be less than 3. For example, 2 + 1 = 3, which would become 0 in GF(3) since 3 is congruent to 0 modulo 3.

Properties of Galois Field

1. Finite Size: The most important property of a Galois Field is that it is finite. It has a specific number of elements, and it is not possible to add any more elements to it. The size of the field is represented by a prime number ‘p’.
2. Closure: The Galois Field remains closed under both addition and multiplication operations, meaning that the result of any operation performed within the set will always be an element of the set.
3. Commutative: The Galois Field is commutative under both addition and multiplication operations. This means that the order of elements does not matter in performing operations. For example, a+b = b+a and ab = ba.
4. Associative: The Galois Field is associative under both addition and multiplication operations. This means that the grouping of elements in an operation does not matter. For example, (a+b) + c = a + (b+c) and (ab) * c = a * (bc).
5. Distributive: The Galois Field follows the distributive property. This means that multiplication distributes over addition, i.e., a * (b + c) = a * b + a * c.
6. Identity Elements: The Galois Field has two identity elements, 0 for addition and 1 for multiplication. Any element added with 0 is equal to the original element, and any element multiplied by 1 is equal to the original element.
7. Inverse Elements: Every element in the Galois Field has an inverse element under both addition and multiplication operations. The inverse element for addition is the negative of the original element, and the inverse element for multiplication is the reciprocal of the original element.

These properties make Galois Fields useful in various fields, such as cryptography, coding theory, and error correction.

Solved Examples – Galois Fields and Its Properties

Example 1: Constructing a Galois Field GF(2)

GF(2) is the simplest non-trivial Galois Field. It consists of two elements: 0 and 1.

Addition table for GF(2):

| 0 1

–+—–

0 | 0 1

1 | 1 0

Multiplication table for GF(2):

| 0 1

–+—–

0 | 0 0

1 | 0 1

Note that addition in GF(2) is equivalent to XOR operation, and multiplication is equivalent to AND operation in binary logic.

Example 2: Constructing GF(4)

GF(4) can be constructed as an extension of GF(2) using the irreducible polynomial x^2 + x + 1.

Elements: {0, 1, α, α+1}, where α is a root of x^2 + x + 1 = 0.

Addition table:

| 0 1 α α+1

—-+——————

0 | 0 1 α α+1

1 | 1 0 α+1 α

α | α α+1 0 1

α+1 | α+1 α 1 0

Multiplication table:

| 0 1 α α+1

—-+——————

0 | 0 0 0 0

1 | 0 1 α α+1

α | 0 α α+1 1

α+1 | 0 α+1 1 α

Example 3: Finding the multiplicative order of elements in GF(7)

GF(7) is a prime field with elements {0, 1, 2, 3, 4, 5, 6}.

Let’s find the multiplicative order of 3:

3^1 ≡ 3 (mod 7)

3^2 ≡ 2 (mod 7)

3^3 ≡ 6 (mod 7)

3^4 ≡ 4 (mod 7)

3^5 ≡ 5 (mod 7)

3^6 ≡ 1 (mod 7)

The multiplicative order of 3 in GF(7) is 6.

Example 4: Finding a primitive element in GF(11)

A primitive element generates all non-zero elements of the field. Let’s check if 2 is primitive in GF(11):

2^1 ≡ 2 (mod 11)

2^2 ≡ 4 (mod 11)

2^3 ≡ 8 (mod 11)

2^4 ≡ 5 (mod 11)

2^5 ≡ 10 (mod 11)

2^6 ≡ 9 (mod 11)

2^7 ≡ 7 (mod 11)

2^8 ≡ 3 (mod 11)

2^9 ≡ 6 (mod 11)

2^10 ≡ 1 (mod 11)

2 generates all non-zero elements of GF(11), so it is a primitive element.

Example 5: Solving a linear equation in GF(5)

Let’s solve the equation 3x + 2 ≡ 4 (mod 5)

Step 1: Subtract 2 from both sides

3x ≡ 2 (mod 5)

Step 2: Multiply both sides by the multiplicative inverse of 3 in GF(5)

3^(-1) ≡ 2 (mod 5) (because 3 * 2 ≡ 1 (mod 5))

So, 2 * 3x ≡ 2 * 2 (mod 5)

x ≡ 4 (mod 5)

Therefore, the solution is x ≡ 4 (mod 5).

Practice Questions

1).Construct the addition and multiplication tables for GF(3).

2).Find a primitive element in GF(13).

3).Solve the equation 4x + 2 ≡ 1 (mod 7) in GF(7).

4).Determine the multiplicative order of 5 in GF(11).

5).Construct GF(8) using the irreducible polynomial x³ + x + 1 over GF(2).

6).Find all generators of the multiplicative group of GF(5).

7).Prove that x² + 1 is irreducible over GF(3).

8).Calculate (x³ + x + 1) * (x² + x) in GF(2)[x]/(x³ + x + 1).

9).Find the minimal polynomial of α^2 + α over GF(2), where α is a root of x³ + x + 1.

10).Determine the number of elements of order 3 in GF(16).

Summary

These practice questions cover a range of topics in Galois Field theory, including field construction, primitive elements, solving linear equations, determining multiplicative orders, irreducibility of polynomials, arithmetic in extension fields, minimal polynomials, and properties of finite field elements. Working through these problems will help reinforce understanding of Galois Field properties, operations, and applications, providing a solid foundation for more advanced concepts in abstract algebra and finite field theory.

FAQs on Galois Fields and Its Properties

What is a Galois Field?

A Galois Field, also known as a finite field, is a mathematical structure that contains a finite number of elements. It satisfies all the axioms of a field, including closure, associativity, commutativity, and distributivity for addition and multiplication, and the existence of additive and multiplicative identities and inverses (except for zero).

What is the difference between GF(p) and GF(pⁿ)?

GF(p), where p is prime, is called a prime field. It contains p elements and is isomorphic to the integers modulo p. GF(pⁿ), where n > 1, is an extension field of GF(p). It contains pⁿ elements and is typically constructed using an irreducible polynomial of degree n over GF(p).

What is a primitive element in a Galois Field?

A primitive element in a Galois Field is an element whose powers generate all non-zero elements of the field. In other words, if α is a primitive element of GF(q), then {α^0, α^1, α^2, …, α^(q-2)} represents all non-zero elements of the field.

How are Galois Fields used in practice?

Galois Fields have numerous practical applications, including:

• Error-correcting codes in digital communications
• Cryptography, particularly in algorithms like AES and elliptic curve cryptography
• Computer algebra systems for efficient computation
• Design of experiments in statistics
• Digital signal processing