Galois Field and Operations

 

What Is a Galois Field?

A Galois Field, or finite field, is a mathematical system with a finite number of elements in which the operations of addition, subtraction, multiplication, and division (except by zero) are well defined and satisfy all field properties.

Galois fields are denoted by GF(pⁿ), where:

• p is a prime number (called the characteristic)

• n is a positive integer

They are named after Évariste Galois, whose work laid the foundation of modern algebra.

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Types of Galois Fields

1. Prime Fields: GF(p)

A prime field contains exactly p elements:

$$GF(p) = \{0, 1, 2, \ldots, p-1\}$$

All arithmetic is performed modulo p.

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Arithmetic Operations in GF(p) (Mod p Arithmetic)

Let us consider GF(7) as an example.

1. Addition (mod p)

Add normally and reduce modulo p.

$$5 + 4 = 9 \equiv 2 \pmod{7}$$

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2. Subtraction (mod p)

Subtract normally and reduce modulo p.

$$3 - 5 = -2 \equiv 5 \pmod{7}$$

Every element has an additive inverse.

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3. Multiplication (mod p)

Multiply normally and reduce modulo p.

$$4 \times 5 = 20 \equiv 6 \pmod{7}$$

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4. Multiplicative Inverse (mod p)

For any non-zero element $a$, there exists an inverse $a^{-1}$ such that:

$$a \cdot a^{-1} \equiv 1 \pmod{p}$$

Example:

$$3^{-1} \equiv 5 \pmod{7}$$

because:

$$3 \times 5 = 15 \equiv 1 \pmod{7}$$

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5. Division (mod p)

Division is multiplication by the inverse:

$$\frac{4}{3} = 4 \times 5 \equiv 6 \pmod{7}$$

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Extension Fields: GF(pⁿ)

When n > 1, the field is an extension field.

Its elements are represented as polynomials of degree less than n, with coefficients from GF(p).

To ensure a valid field, polynomial arithmetic is reduced using an irreducible polynomial of degree n.

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Irreducible Polynomials

An irreducible polynomial is one that cannot be factored over a given field.

It plays the same role as a prime number in integer arithmetic.

Example (over GF(2)):

$$x^3 + x + 1$$

This polynomial is irreducible and is commonly used to construct GF(2³).

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Representation of Elements in GF(2³)

Consider:

$$GF(2^3) \text{ with irreducible polynomial } x^3 + x + 1$$

Each element is a polynomial of degree at most 2:

$$a_2x^2 + a_1x + a_0,\quad a_i \in \{0,1\}$$

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Arithmetic Operations in Extension Fields

1. Addition

• Coefficients are added modulo 2

• No carry is allowed

Example:

$$(x^2 + x + 1) + (x^2 + 1) = x$$

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2. Subtraction

In GF(2), subtraction is identical to addition.

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3. Multiplication

Step 1: Multiply polynomials normally
Step 2: Reduce using the irreducible polynomial

Example:

$$(x + 1)(x^2 + 1) = x^3 + x^2 + x + 1$$

Since:

$$x^3 \equiv x + 1$$
$$\Rightarrow x^2$$

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4. Multiplicative Inverse

The inverse of $a(x)$ satisfies:

$$a(x)\cdot a(x)^{-1} \equiv 1$$

Example:

$$(x + 1)^{-1} = x^2 + x$$

because:

$$(x + 1)(x^2 + x) \equiv 1$$

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5. Division

$$\frac{a(x)}{b(x)} = a(x)\cdot b(x)^{-1}$$

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6. Exponentiation

Exponentiation is repeated multiplication with reduction.

Example:

$$(x + 1)^3 \equiv x^2$$

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Why Galois Fields Matter

Galois fields are fundamental in:

• Cryptography (AES uses GF(2⁸))

• Error-correcting codes

• Digital communication

• Computer algebra systems

• Number-theoretic algorithms

• AKS algorithm

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Conclusion

Galois fields unify modular arithmetic and polynomial algebra into a powerful framework for computation over finite sets. Prime fields use integer arithmetic modulo p, while extension fields use polynomial arithmetic modulo an irreducible polynomial, enabling applications across modern mathematics and computer science.