GF(2)
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GF(2) is the Galois field (or finite field) of two elements. The two elements are nearly always called 0 and 1. The field's addition operation is given by the table
| + | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
and its multiplication operation by the following table.
| × | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
These two elements and these two operations constitute a system with many of the important properties of familiar number systems: addition and multiplication are commutative and associative, multiplication is distributive over addition, addition has an identity element (0) and an inverse for every element, and multiplication has an identity element (1) and an inverse for every element but 0.
Because of these properties, many familiar and powerful tools of mathematics work in GF(2) just as well as in the real numbers. For example, many techniques of matrix algebra work on matrices whose elements are in GF(2), including matrix inversion, which is important in the analysis of many binary algorithms.
