GF(2)
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GF(2) (also F2, Z/2Z or Z2) is the Galois field of two elements. It is the smallest finite field.
Definition
The two elements are nearly always called 0 and 1, being the additive and multiplicative identities, respectively.
The field's addition operation is given by the table below, which corresponds to the logical XOR operation.
| + | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
The field's multiplication operation corresponds to the logical AND operation.
| × | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
One may also define GF(2) as 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;
- every element x of GF(2) satisfies x2 = x.
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 abelian group V with the property v+v=0 for every v in V 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).
Since modern computers represent data with binary code, a field with two elements, GF(2), is an important tool for studying algorithms on these machines. GF(2) can be extended to arbitrarily large fields GF(2n), allowing definition of bitwise operations on strings of bits. Properties of LFSRs, checksums and some ciphers can be studied in this way.
See also
References
- Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069.