Galois Field is a finite #Field, meaning it contains a finite number of elements which mathematical operations are defined and satisfy certain basic rules. We can do addition, subtraction, multiplication, and division without leaving the field. The total number of elements in Galois Field must be \(p^n\) where \(p\) is a prime number. It is denoted as \(GF(p^n)\).
Common in Cryptography#, \(GF(p)\) is a Set# where \(\{a \in P | a \in \mathbb{Z} \text{ and } 0 \le a \le p - 1 \}\) with modulo# prime \(p\) as its arithmetic operations. For example, \(GF(7)\) multiplication looks like the table below.
| \(\times\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 0 | 2 | 4 | 6 | 1 | 3 | 5 |
| 3 | 0 | 3 | 6 | 2 | 5 | 1 | 4 |
| 4 | 0 | 4 | 1 | 5 | 2 | 6 | 3 |
| 5 | 0 | 5 | 3 | 1 | 6 | 4 | 2 |
| 6 | 0 | 6 | 5 | 4 | 3 | 2 | 1 |
Note: Notice the #multiplicative inverse nature of the Galois Field.
Note: The #inverse of the \(a\) could be found in the table by finding the matched row or column with value 1. For example, in the column \(4\), the value 1 appear in the row \(2\), thus \(2\) is the inverse of \(4\) in modulo 7.