What is a Galois field?

A Galois field, also known as a finite field, is a mathematical structure with a finite number of elements.

Note: The Galois field is named after Évariste Galois.

Data can be treated as vectors in a Galois field. When we treat data as vectors in a Galois field, we consider each data element as a vector component within the Galois field. In the case of computer data, where binary representation is common, we can view each bit (0 or 1) as an element of the Galois field.

Mathematical explanation

Mathematically, a Galois field of order $q$, denoted as $GF(q)$, is defined as a set of $q$ elements ${0, 1, 2, ..., q-1}$.

An example of a Galois field is $GF(2)$, also known as the binary field or the field of two elements. In $GF(2)$, there are two elements, $0$ and $1$, and the addition and multiplication operations follow specific rules:

• Addition in $GF(2)$:
  $0 + 0 = 0$
  $0 + 1 = 1$
  $1 + 0 = 1$
  $1 + 1 = 0$

• Multiplication in $GF(2)$:
  $0 * 0 = 0$
  $0 * 1 = 0$
  $1 * 0 = 0$
  $1 * 1 = 1$

In $GF(2)$, the addition operation is equivalent to the XOR (exclusive OR) operation, while the multiplication operation is equivalent to the AND operation.

Another instance of a Galois field is $GF(3)$, comprising three elements: $0$, $1$, and $2$. In this field, the addition and multiplication operations are executed modulo $3$, ensuring that the result of any operation remains within the range of $0$ to $2$. For instance, when adding $2$ and $1$, the result is $3$. However, in $GF(3)$, this value is reduced to $0$ since $3$ is congruent to $0$ modulo $3$.

Applications of Galois fields

There are various applications of Galois fields, such as:

• Cryptography

• Coding theory

• Scrambling data