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Concept
GF(2)
Summary
(also denoted , Z/2Z or ) is the finite field of two elements (GF is the initialism of Galois field, another name for finite fields). Notations Z_2 and may be encountered although they can be confused with the notation of 2-adic integers.
GF(2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual.
The elements of GF(2) may be identified with the two possible values of a bit and to the boolean values true and false. It follows that GF(2) is fundamental and ubiquitous in computer science and its logical foundations.
GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.
Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below:
If the elements of GF(2) are seen as boolean values, then the addition is the same as that of the logical XOR operation.
Since each element equals its opposite, subtraction is thus the same operation as addition.
The multiplication of GF(2) is again the usual multiplication modulo 2 (see the table below), and on boolean variables corresponds to the logical AND operation.
GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z.
finite field
Because GF(2) is a field, many of the familiar properties of number systems such as the rational numbers and real numbers are retained:
addition has an identity element (0) and an inverse for every element;
multiplication has an identity element (1) and an inverse for every element but 0;
addition and multiplication are commutative and associative;
multiplication is distributive over addition.
Properties that are not familiar from the real numbers include:
every element x of GF(2) satisfies x + x = 0 and therefore −x = x; this means that the characteristic of GF(2) is 2;
every element x of GF(2) satisfies x2 = x (i.
Official source
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