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 qq, denoted as GF(q)GF(q), is defined as a set of qq elements 0,1,2,...,q1{0, 1, 2, ..., q-1}.

An example of a Galois field is GF(2)GF(2), also known as the binary field or the field of two elements. In GF(2)GF(2), there are two elements, 00 and 11, and the addition and multiplication operations follow specific rules:

  • Addition in GF(2)GF(2):
    0+0=00 + 0 = 0
    0+1=10 + 1 = 1
    1+0=11 + 0 = 1
    1+1=01 + 1 = 0

  • Multiplication in GF(2)GF(2):
    00=00 * 0 = 0
    01=00 * 1 = 0
    10=01 * 0 = 0
    11=11 * 1 = 1

In GF(2)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)GF(3), comprising three elements: 00, 11, and 22. In this field, the addition and multiplication operations are executed modulo 33, ensuring that the result of any operation remains within the range of 00 to 22. For instance, when adding 22 and 11, the result is 33. However, in GF(3)GF(3), this value is reduced to 00 since 33 is congruent to 00 modulo 33.

Applications of Galois fields

There are various applications of Galois fields, such as:

  • Cryptography

  • Coding theory

  • Scrambling data

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