GF(2)
GF(2) (also denoted , Z/2Z or ) is the Galois field of two elements (GF is the initialism of "Galois field"). Notations Z2 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. It follows that GF(2) is fundamental and ubiquitous in computer science.
Definition
GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.
Its addition is defined by the table below, which is the same as that of the logical XOR operation.
| + | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Each element equals its opposite, and subtraction is thus the same operation as addition.
The multiplication of GF(2) is defined by the table below, which is the same as that of the logical AND operation.
| × | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
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.
Properties
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.e. is idempotent with respect to multiplication); this is an instance of Fermat's little theorem. GF(2) is the only field with this property (Proof: if , then either or . In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives . All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property).
Applications
Because of the algebraic properties above, many familiar and powerful tools of mathematics work in GF(2) just as well as other fields. For example, matrix operations, including matrix inversion, can be applied to matrices with elements in GF(2) (see matrix ring).
Any group V with the property v + v = 0 for every v in V (i.e. every element is an involution) is necessarily abelian and can be turned into a vector space over GF(2) in a natural fashion, by defining 0v = 0 and 1v = v. This vector space will have a basis, implying that the number of elements of V must be a power of 2 (or infinite).
In modern computers, data are represented with bit strings of a fixed length, called machine words. These are endowed with the structure of a vector space over GF(2). The addition of this vector space is the bitwise operation called XOR (exclusive or). The bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science. These spaces can also be augmented with a multiplication operation that makes them into a field GF(2n), but the multiplication operation cannot be a bitwise operation. When n is itself a power of two, the multiplication operation can be nim-multiplication; alternatively, for any n, one can use multiplication of polynomials over GF(2) modulo a irreducible polynomial (as for instance for the field GF(28) in the description of the Advanced Encryption Standard cipher).
See also
References
- Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069.