Addition and Subtraction
Addition and subtraction are performed by adding or subtracting two of these polynomials together, and reducing the result modulo the characteristic.
In a finite field with characteristic 2, addition modulo 2, subtraction modulo 2, and XOR are identical. Thus,
- Polynomial: (x6 + x4 + x + 1) + (x7 + x6 + x3 + x) = x7 + x4 + x3 + 1
- Binary: {01010011} + {11001010} = {10011001}
- Hexadecimal: {53} + {CA} = {99}
Notice that under regular addition of polynomials, the sum would contain a term 2x6, but that this term becomes 0x6 and is dropped when the answer is reduced modulo 2.
Here is a table with both the normal algebraic sum and the characteristic 2 finite field sum of a few polynomials:
| p1 | p2 | p1 + p2 (normal algebra) | p1 + p2 in GF(2n) |
|---|---|---|---|
| x3 + x + 1 | x3 + x2 | 2x3 + x2 + x + 1 | x2 + x + 1 |
| x4 + x2 | x6 + x2 | x6 + x4 + 2x2 | x6 + x4 |
| x + 1 | x2 + 1 | x2 + x + 2 | x2 + x |
| x3 + x | x2 + 1 | x3 + x2 + x + 1 | x3 + x2 + x + 1 |
| x2 + x | x2 + x | 2x2 + 2x | 0 |
Note: In computer science applications, the operations are simplified for finite fields of characteristic 2, also called GF(2n) Galois fields, making these fields especially popular choices for applications.
Read more about this topic: Finite Field Arithmetic
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