ECE4253 Digital Communications | |
Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada | |
This page presents addition and multiplication tables for Galois fields GF(2m).
Addition Table
Values in GF(25) are 5-bits each, spanning the decimal range [0..31]. Addition takes place on these 5-bit binary values using bitwise XOR.
For example: 8 + 4 = (01000) + (00100) = (01100) = 12 (highlighted below)
The choice of polynomial P(x) plays no role in the addition operation.
+ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
1 | 1 | 0 | 3 | 2 | 5 | 4 | 7 | 6 | 9 | 8 | 11 | 10 | 13 | 12 | 15 | 14 | 17 | 16 | 19 | 18 | 21 | 20 | 23 | 22 | 25 | 24 | 27 | 26 | 29 | 28 | 31 | 30 |
2 | 2 | 3 | 0 | 1 | 6 | 7 | 4 | 5 | 10 | 11 | 8 | 9 | 14 | 15 | 12 | 13 | 18 | 19 | 16 | 17 | 22 | 23 | 20 | 21 | 26 | 27 | 24 | 25 | 30 | 31 | 28 | 29 |
3 | 3 | 2 | 1 | 0 | 7 | 6 | 5 | 4 | 11 | 10 | 9 | 8 | 15 | 14 | 13 | 12 | 19 | 18 | 17 | 16 | 23 | 22 | 21 | 20 | 27 | 26 | 25 | 24 | 31 | 30 | 29 | 28 |
4 | 4 | 5 | 6 | 7 | 0 | 1 | 2 | 3 | 12 | 13 | 14 | 15 | 8 | 9 | 10 | 11 | 20 | 21 | 22 | 23 | 16 | 17 | 18 | 19 | 28 | 29 | 30 | 31 | 24 | 25 | 26 | 27 |
5 | 5 | 4 | 7 | 6 | 1 | 0 | 3 | 2 | 13 | 12 | 15 | 14 | 9 | 8 | 11 | 10 | 21 | 20 | 23 | 22 | 17 | 16 | 19 | 18 | 29 | 28 | 31 | 30 | 25 | 24 | 27 | 26 |
6 | 6 | 7 | 4 | 5 | 2 | 3 | 0 | 1 | 14 | 15 | 12 | 13 | 10 | 11 | 8 | 9 | 22 | 23 | 20 | 21 | 18 | 19 | 16 | 17 | 30 | 31 | 28 | 29 | 26 | 27 | 24 | 25 |
7 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 |
8 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
9 | 9 | 8 | 11 | 10 | 13 | 12 | 15 | 14 | 1 | 0 | 3 | 2 | 5 | 4 | 7 | 6 | 25 | 24 | 27 | 26 | 29 | 28 | 31 | 30 | 17 | 16 | 19 | 18 | 21 | 20 | 23 | 22 |
10 | 10 | 11 | 8 | 9 | 14 | 15 | 12 | 13 | 2 | 3 | 0 | 1 | 6 | 7 | 4 | 5 | 26 | 27 | 24 | 25 | 30 | 31 | 28 | 29 | 18 | 19 | 16 | 17 | 22 | 23 | 20 | 21 |
11 | 11 | 10 | 9 | 8 | 15 | 14 | 13 | 12 | 3 | 2 | 1 | 0 | 7 | 6 | 5 | 4 | 27 | 26 | 25 | 24 | 31 | 30 | 29 | 28 | 19 | 18 | 17 | 16 | 23 | 22 | 21 | 20 |
12 | 12 | 13 | 14 | 15 | 8 | 9 | 10 | 11 | 4 | 5 | 6 | 7 | 0 | 1 | 2 | 3 | 28 | 29 | 30 | 31 | 24 | 25 | 26 | 27 | 20 | 21 | 22 | 23 | 16 | 17 | 18 | 19 |
13 | 13 | 12 | 15 | 14 | 9 | 8 | 11 | 10 | 5 | 4 | 7 | 6 | 1 | 0 | 3 | 2 | 29 | 28 | 31 | 30 | 25 | 24 | 27 | 26 | 21 | 20 | 23 | 22 | 17 | 16 | 19 | 18 |
14 | 14 | 15 | 12 | 13 | 10 | 11 | 8 | 9 | 6 | 7 | 4 | 5 | 2 | 3 | 0 | 1 | 30 | 31 | 28 | 29 | 26 | 27 | 24 | 25 | 22 | 23 | 20 | 21 | 18 | 19 | 16 | 17 |
15 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 |
16 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
17 | 17 | 16 | 19 | 18 | 21 | 20 | 23 | 22 | 25 | 24 | 27 | 26 | 29 | 28 | 31 | 30 | 1 | 0 | 3 | 2 | 5 | 4 | 7 | 6 | 9 | 8 | 11 | 10 | 13 | 12 | 15 | 14 |
18 | 18 | 19 | 16 | 17 | 22 | 23 | 20 | 21 | 26 | 27 | 24 | 25 | 30 | 31 | 28 | 29 | 2 | 3 | 0 | 1 | 6 | 7 | 4 | 5 | 10 | 11 | 8 | 9 | 14 | 15 | 12 | 13 |
19 | 19 | 18 | 17 | 16 | 23 | 22 | 21 | 20 | 27 | 26 | 25 | 24 | 31 | 30 | 29 | 28 | 3 | 2 | 1 | 0 | 7 | 6 | 5 | 4 | 11 | 10 | 9 | 8 | 15 | 14 | 13 | 12 |
20 | 20 | 21 | 22 | 23 | 16 | 17 | 18 | 19 | 28 | 29 | 30 | 31 | 24 | 25 | 26 | 27 | 4 | 5 | 6 | 7 | 0 | 1 | 2 | 3 | 12 | 13 | 14 | 15 | 8 | 9 | 10 | 11 |
21 | 21 | 20 | 23 | 22 | 17 | 16 | 19 | 18 | 29 | 28 | 31 | 30 | 25 | 24 | 27 | 26 | 5 | 4 | 7 | 6 | 1 | 0 | 3 | 2 | 13 | 12 | 15 | 14 | 9 | 8 | 11 | 10 |
22 | 22 | 23 | 20 | 21 | 18 | 19 | 16 | 17 | 30 | 31 | 28 | 29 | 26 | 27 | 24 | 25 | 6 | 7 | 4 | 5 | 2 | 3 | 0 | 1 | 14 | 15 | 12 | 13 | 10 | 11 | 8 | 9 |
23 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 |
24 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
25 | 25 | 24 | 27 | 26 | 29 | 28 | 31 | 30 | 17 | 16 | 19 | 18 | 21 | 20 | 23 | 22 | 9 | 8 | 11 | 10 | 13 | 12 | 15 | 14 | 1 | 0 | 3 | 2 | 5 | 4 | 7 | 6 |
26 | 26 | 27 | 24 | 25 | 30 | 31 | 28 | 29 | 18 | 19 | 16 | 17 | 22 | 23 | 20 | 21 | 10 | 11 | 8 | 9 | 14 | 15 | 12 | 13 | 2 | 3 | 0 | 1 | 6 | 7 | 4 | 5 |
27 | 27 | 26 | 25 | 24 | 31 | 30 | 29 | 28 | 19 | 18 | 17 | 16 | 23 | 22 | 21 | 20 | 11 | 10 | 9 | 8 | 15 | 14 | 13 | 12 | 3 | 2 | 1 | 0 | 7 | 6 | 5 | 4 |
28 | 28 | 29 | 30 | 31 | 24 | 25 | 26 | 27 | 20 | 21 | 22 | 23 | 16 | 17 | 18 | 19 | 12 | 13 | 14 | 15 | 8 | 9 | 10 | 11 | 4 | 5 | 6 | 7 | 0 | 1 | 2 | 3 |
29 | 29 | 28 | 31 | 30 | 25 | 24 | 27 | 26 | 21 | 20 | 23 | 22 | 17 | 16 | 19 | 18 | 13 | 12 | 15 | 14 | 9 | 8 | 11 | 10 | 5 | 4 | 7 | 6 | 1 | 0 | 3 | 2 |
30 | 30 | 31 | 28 | 29 | 26 | 27 | 24 | 25 | 22 | 23 | 20 | 21 | 18 | 19 | 16 | 17 | 14 | 15 | 12 | 13 | 10 | 11 | 8 | 9 | 6 | 7 | 4 | 5 | 2 | 3 | 0 | 1 |
31 | 31 | 30 | 29 | 28 | 27 | 26 | 25 | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
Multiplication Table
Values in GF(25) are 5-bits each, spanning the decimal range [0..31]. Multiplication takes place on 5-bit binary values (with modulo 2 addition) and then the result is computed modulo P(x) = (100101) = 37 (decimal).
For example: 11 × 4 = (01011) × (00100) = (101100) = (01001) mod (100101) = 9 (highlighted below)
The specific polynomial P(x) provides the modulus for the multiplication results.
× | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | 26 | 28 | 30 | 5 | 7 | 1 | 3 | 13 | 15 | 9 | 11 | 21 | 23 | 17 | 19 | 29 | 31 | 25 | 27 |
3 | 3 | 6 | 5 | 12 | 15 | 10 | 9 | 24 | 27 | 30 | 29 | 20 | 23 | 18 | 17 | 21 | 22 | 19 | 16 | 25 | 26 | 31 | 28 | 13 | 14 | 11 | 8 | 1 | 2 | 7 | 4 |
4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 5 | 1 | 13 | 9 | 21 | 17 | 29 | 25 | 10 | 14 | 2 | 6 | 26 | 30 | 18 | 22 | 15 | 11 | 7 | 3 | 31 | 27 | 23 | 19 |
5 | 5 | 10 | 15 | 20 | 17 | 30 | 27 | 13 | 8 | 7 | 2 | 25 | 28 | 19 | 22 | 26 | 31 | 16 | 21 | 14 | 11 | 4 | 1 | 23 | 18 | 29 | 24 | 3 | 6 | 9 | 12 |
6 | 6 | 12 | 10 | 24 | 30 | 20 | 18 | 21 | 19 | 25 | 31 | 13 | 11 | 1 | 7 | 15 | 9 | 3 | 5 | 23 | 17 | 27 | 29 | 26 | 28 | 22 | 16 | 2 | 4 | 14 | 8 |
7 | 7 | 14 | 9 | 28 | 27 | 18 | 21 | 29 | 26 | 19 | 20 | 1 | 6 | 15 | 8 | 31 | 24 | 17 | 22 | 3 | 4 | 13 | 10 | 2 | 5 | 12 | 11 | 30 | 25 | 16 | 23 |
8 | 8 | 16 | 24 | 5 | 13 | 21 | 29 | 10 | 2 | 26 | 18 | 15 | 7 | 31 | 23 | 20 | 28 | 4 | 12 | 17 | 25 | 1 | 9 | 30 | 22 | 14 | 6 | 27 | 19 | 11 | 3 |
9 | 9 | 18 | 27 | 1 | 8 | 19 | 26 | 2 | 11 | 16 | 25 | 3 | 10 | 17 | 24 | 4 | 13 | 22 | 31 | 5 | 12 | 23 | 30 | 6 | 15 | 20 | 29 | 7 | 14 | 21 | 28 |
10 | 10 | 20 | 30 | 13 | 7 | 25 | 19 | 26 | 16 | 14 | 4 | 23 | 29 | 3 | 9 | 17 | 27 | 5 | 15 | 28 | 22 | 8 | 2 | 11 | 1 | 31 | 21 | 6 | 12 | 18 | 24 |
11 | 11 | 22 | 29 | 9 | 2 | 31 | 20 | 18 | 25 | 4 | 15 | 27 | 16 | 13 | 6 | 1 | 10 | 23 | 28 | 8 | 3 | 30 | 21 | 19 | 24 | 5 | 14 | 26 | 17 | 12 | 7 |
12 | 12 | 24 | 20 | 21 | 25 | 13 | 1 | 15 | 3 | 23 | 27 | 26 | 22 | 2 | 14 | 30 | 18 | 6 | 10 | 11 | 7 | 19 | 31 | 17 | 29 | 9 | 5 | 4 | 8 | 28 | 16 |
13 | 13 | 26 | 23 | 17 | 28 | 11 | 6 | 7 | 10 | 29 | 16 | 22 | 27 | 12 | 1 | 14 | 3 | 20 | 25 | 31 | 18 | 5 | 8 | 9 | 4 | 19 | 30 | 24 | 21 | 2 | 15 |
14 | 14 | 28 | 18 | 29 | 19 | 1 | 15 | 31 | 17 | 3 | 13 | 2 | 12 | 30 | 16 | 27 | 21 | 7 | 9 | 6 | 8 | 26 | 20 | 4 | 10 | 24 | 22 | 25 | 23 | 5 | 11 |
15 | 15 | 30 | 17 | 25 | 22 | 7 | 8 | 23 | 24 | 9 | 6 | 14 | 1 | 16 | 31 | 11 | 4 | 21 | 26 | 18 | 29 | 12 | 3 | 28 | 19 | 2 | 13 | 5 | 10 | 27 | 20 |
16 | 16 | 5 | 21 | 10 | 26 | 15 | 31 | 20 | 4 | 17 | 1 | 30 | 14 | 27 | 11 | 13 | 29 | 8 | 24 | 7 | 23 | 2 | 18 | 25 | 9 | 28 | 12 | 19 | 3 | 22 | 6 |
17 | 17 | 7 | 22 | 14 | 31 | 9 | 24 | 28 | 13 | 27 | 10 | 18 | 3 | 21 | 4 | 29 | 12 | 26 | 11 | 19 | 2 | 20 | 5 | 1 | 16 | 6 | 23 | 15 | 30 | 8 | 25 |
18 | 18 | 1 | 19 | 2 | 16 | 3 | 17 | 4 | 22 | 5 | 23 | 6 | 20 | 7 | 21 | 8 | 26 | 9 | 27 | 10 | 24 | 11 | 25 | 12 | 30 | 13 | 31 | 14 | 28 | 15 | 29 |
19 | 19 | 3 | 16 | 6 | 21 | 5 | 22 | 12 | 31 | 15 | 28 | 10 | 25 | 9 | 26 | 24 | 11 | 27 | 8 | 30 | 13 | 29 | 14 | 20 | 7 | 23 | 4 | 18 | 1 | 17 | 2 |
20 | 20 | 13 | 25 | 26 | 14 | 23 | 3 | 17 | 5 | 28 | 8 | 11 | 31 | 6 | 18 | 7 | 19 | 10 | 30 | 29 | 9 | 16 | 4 | 22 | 2 | 27 | 15 | 12 | 24 | 1 | 21 |
21 | 21 | 15 | 26 | 30 | 11 | 17 | 4 | 25 | 12 | 22 | 3 | 7 | 18 | 8 | 29 | 23 | 2 | 24 | 13 | 9 | 28 | 6 | 19 | 14 | 27 | 1 | 20 | 16 | 5 | 31 | 10 |
22 | 22 | 9 | 31 | 18 | 4 | 27 | 13 | 1 | 23 | 8 | 30 | 19 | 5 | 26 | 12 | 2 | 20 | 11 | 29 | 16 | 6 | 25 | 15 | 3 | 21 | 10 | 28 | 17 | 7 | 24 | 14 |
23 | 23 | 11 | 28 | 22 | 1 | 29 | 10 | 9 | 30 | 2 | 21 | 31 | 8 | 20 | 3 | 18 | 5 | 25 | 14 | 4 | 19 | 15 | 24 | 27 | 12 | 16 | 7 | 13 | 26 | 6 | 17 |
24 | 24 | 21 | 13 | 15 | 23 | 26 | 2 | 30 | 6 | 11 | 19 | 17 | 9 | 4 | 28 | 25 | 1 | 12 | 20 | 22 | 14 | 3 | 27 | 7 | 31 | 18 | 10 | 8 | 16 | 29 | 5 |
25 | 25 | 23 | 14 | 11 | 18 | 28 | 5 | 22 | 15 | 1 | 24 | 29 | 4 | 10 | 19 | 9 | 16 | 30 | 7 | 2 | 27 | 21 | 12 | 31 | 6 | 8 | 17 | 20 | 13 | 3 | 26 |
26 | 26 | 17 | 11 | 7 | 29 | 22 | 12 | 14 | 20 | 31 | 5 | 9 | 19 | 24 | 2 | 28 | 6 | 13 | 23 | 27 | 1 | 10 | 16 | 18 | 8 | 3 | 25 | 21 | 15 | 4 | 30 |
27 | 27 | 19 | 8 | 3 | 24 | 16 | 11 | 6 | 29 | 21 | 14 | 5 | 30 | 22 | 13 | 12 | 23 | 31 | 4 | 15 | 20 | 28 | 7 | 10 | 17 | 25 | 2 | 9 | 18 | 26 | 1 |
28 | 28 | 29 | 1 | 31 | 3 | 2 | 30 | 27 | 7 | 6 | 26 | 4 | 24 | 25 | 5 | 19 | 15 | 14 | 18 | 12 | 16 | 17 | 13 | 8 | 20 | 21 | 9 | 23 | 11 | 10 | 22 |
29 | 29 | 31 | 2 | 27 | 6 | 4 | 25 | 19 | 14 | 12 | 17 | 8 | 21 | 23 | 10 | 3 | 30 | 28 | 1 | 24 | 5 | 7 | 26 | 16 | 13 | 15 | 18 | 11 | 22 | 20 | 9 |
30 | 30 | 25 | 7 | 23 | 9 | 14 | 16 | 11 | 21 | 18 | 12 | 28 | 2 | 5 | 27 | 22 | 8 | 15 | 17 | 1 | 31 | 24 | 6 | 29 | 3 | 4 | 26 | 10 | 20 | 19 | 13 |
31 | 31 | 27 | 4 | 19 | 12 | 8 | 23 | 3 | 28 | 24 | 7 | 16 | 15 | 11 | 20 | 6 | 25 | 29 | 2 | 21 | 10 | 14 | 17 | 5 | 26 | 30 | 1 | 22 | 9 | 13 | 18 |
Select a primitive polynomial P(x)
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2024-04-26 08:58:00 ADT
Last Updated: 2011-02-02 |
Richard Tervo [ tervo@unb.ca ] | Back to the course homepage... |