This page implements Galois Fields GF(\(p^n\)) with Python and provides a table of multiplication. With this \(p\) is a prime number and \(n\) ranges from 1 to \(n-1\). A GF(\(2^4\)) will have a range of outputs from 0 to 15. In this case, we will determine the Additive, Substractive and Multiplicative Group Table for for GF(\(2^3\)), GF(\(3^2\)), GF(\(5^2\)), and a range of other values.
Additive, Division and Multiplicative Group Table for GF(\(p^n\)) for a range of \(p\) and \(n\) values |
Coding
The coding is:
import galois import sys p=2 n=2 if (len(sys.argv)>1): p=int(sys.argv[1]) if (len(sys.argv)>2): n=int(sys.argv[2]) print(f"== G({p}^{n}) ==\n") GF = galois.GF(p**n) print("\n",GF.arithmetic_table("+")) GF = galois.GF(p**n) print("\n",GF.arithmetic_table("*")) GF = galois.GF(p**n) print("\n",GF.arithmetic_table("/"))
With GF(\(2^2\)):
x + y | 0 1 2 3 ------|------------ 0 | 0 1 2 3 1 | 1 0 3 2 2 | 2 3 0 1 3 | 3 2 1 0 x * y | 0 1 2 3 ------|------------ 0 | 0 0 0 0 1 | 0 1 2 3 2 | 0 2 3 1 3 | 0 3 1 2 x / y | 1 2 3 ------|--------- 0 | 0 0 0 1 | 1 3 2 2 | 2 1 3 3 | 3 2 1
With GF(\(2^3\)):
x + y | 0 1 2 3 4 5 6 7 ------|------------------------ 0 | 0 1 2 3 4 5 6 7 1 | 1 0 3 2 5 4 7 6 2 | 2 3 0 1 6 7 4 5 3 | 3 2 1 0 7 6 5 4 4 | 4 5 6 7 0 1 2 3 5 | 5 4 7 6 1 0 3 2 6 | 6 7 4 5 2 3 0 1 7 | 7 6 5 4 3 2 1 0 x * y | 0 1 2 3 4 5 6 7 ------|------------------------ 0 | 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 2 | 0 2 4 6 3 1 7 5 3 | 0 3 6 5 7 4 1 2 4 | 0 4 3 7 6 2 5 1 5 | 0 5 1 4 2 7 3 6 6 | 0 6 7 1 5 3 2 4 7 | 0 7 5 2 1 6 4 3 x / y | 1 2 3 4 5 6 7 ------|--------------------- 0 | 0 0 0 0 0 0 0 1 | 1 5 6 7 2 3 4 2 | 2 1 7 5 4 6 3 3 | 3 4 1 2 6 5 7 4 | 4 2 5 1 3 7 6 5 | 5 7 3 6 1 4 2 6 | 6 3 2 4 7 1 5 7 | 7 6 4 3 5 2 1
With GF(\(2^4\)):
x + y | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------|---------------------------------------------------------------- 0 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 1 0 3 2 5 4 7 6 9 8 11 10 13 12 15 14 2 | 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 | 3 2 1 0 7 6 5 4 11 10 9 8 15 14 13 12 4 | 4 5 6 7 0 1 2 3 12 13 14 15 8 9 10 11 5 | 5 4 7 6 1 0 3 2 13 12 15 14 9 8 11 10 6 | 6 7 4 5 2 3 0 1 14 15 12 13 10 11 8 9 7 | 7 6 5 4 3 2 1 0 15 14 13 12 11 10 9 8 8 | 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 | 9 8 11 10 13 12 15 14 1 0 3 2 5 4 7 6 10 | 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 | 11 10 9 8 15 14 13 12 3 2 1 0 7 6 5 4 12 | 12 13 14 15 8 9 10 11 4 5 6 7 0 1 2 3 13 | 13 12 15 14 9 8 11 10 5 4 7 6 1 0 3 2 14 | 14 15 12 13 10 11 8 9 6 7 4 5 2 3 0 1 15 | 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 x * y | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------|---------------------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2 | 0 2 4 6 8 10 12 14 3 1 7 5 11 9 15 13 3 | 0 3 6 5 12 15 10 9 11 8 13 14 7 4 1 2 4 | 0 4 8 12 3 7 11 15 6 2 14 10 5 1 13 9 5 | 0 5 10 15 7 2 13 8 14 11 4 1 9 12 3 6 6 | 0 6 12 10 11 13 7 1 5 3 9 15 14 8 2 4 7 | 0 7 14 9 15 8 1 6 13 10 3 4 2 5 12 11 8 | 0 8 3 11 6 14 5 13 12 4 15 7 10 2 9 1 9 | 0 9 1 8 2 11 3 10 4 13 5 12 6 15 7 14 10 | 0 10 7 13 14 4 9 3 15 5 8 2 1 11 6 12 11 | 0 11 5 14 10 1 15 4 7 12 2 9 13 6 8 3 12 | 0 12 11 7 5 9 14 2 10 6 1 13 15 3 4 8 13 | 0 13 9 4 1 12 8 5 2 15 11 6 3 14 10 7 14 | 0 14 15 1 13 3 2 12 9 7 6 8 4 10 11 5 15 | 0 15 13 2 9 6 4 11 1 14 12 3 8 7 5 10 x / y | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------|------------------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 1 9 14 13 11 7 6 15 2 12 5 10 4 3 8 2 | 2 1 15 9 5 14 12 13 4 11 10 7 8 6 3 3 | 3 8 1 4 14 9 10 2 6 7 15 13 12 5 11 4 | 4 2 13 1 10 15 11 9 8 5 7 14 3 12 6 5 | 5 11 3 12 1 8 13 6 10 9 2 4 7 15 14 6 | 6 3 2 8 15 1 7 4 12 14 13 9 11 10 5 7 | 7 10 12 5 4 6 1 11 14 2 8 3 15 9 13 8 | 8 4 9 2 7 13 5 1 3 10 14 15 6 11 12 9 | 9 13 7 15 12 10 3 14 1 6 11 5 2 8 4 10 | 10 5 6 11 2 3 9 12 7 1 4 8 14 13 15 11 | 11 12 8 6 9 4 15 3 5 13 1 2 10 14 7 12 | 12 6 4 3 13 2 14 8 11 15 9 1 5 7 10 13 | 13 15 10 14 6 5 8 7 9 3 12 11 1 4 2 14 | 14 7 11 10 8 12 2 5 15 4 3 6 13 1 9 15 | 15 14 5 7 3 11 4 10 13 8 6 12 9 2 1
With GF(\(2^5\)):
== G(2^5) == x + y | 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 x * y | 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 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 2 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 | 0 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 x / y | 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 1 18 28 9 23 14 12 22 4 25 16 7 15 6 13 11 24 2 29 30 26 8 5 17 10 21 31 3 19 20 27 2 | 2 1 29 18 11 28 24 9 8 23 5 14 30 12 26 22 21 4 31 25 17 16 10 7 20 15 27 6 3 13 19 3 | 3 19 1 27 28 18 20 31 12 14 21 9 17 10 23 29 13 6 2 7 11 24 15 22 30 26 4 5 16 25 8 4 | 4 2 31 1 22 29 21 18 16 11 10 28 25 24 17 9 15 8 27 23 7 5 20 14 13 30 19 12 6 26 3 5 | 5 16 3 8 1 19 25 4 20 18 26 27 22 30 28 2 23 10 6 9 29 13 17 31 7 11 12 15 21 14 24 6 | 6 3 2 19 29 1 13 27 24 28 15 18 7 20 11 31 26 12 4 14 22 21 30 9 25 17 8 10 5 23 16 7 | 7 17 30 26 10 15 1 13 28 5 31 21 8 18 6 20 2 14 25 16 12 29 27 24 19 4 23 9 22 3 11 8 | 8 4 27 2 9 31 15 1 5 22 20 29 23 21 7 18 30 16 19 11 14 10 13 28 26 25 3 24 12 17 6 9 | 9 22 7 11 30 17 3 23 1 15 4 26 24 19 10 25 6 18 14 21 20 2 8 13 16 12 28 27 31 5 29 10 | 10 5 6 16 2 3 23 8 13 1 17 19 9 25 29 4 11 20 12 18 31 26 7 27 14 22 24 30 15 28 21 11 | 11 23 26 25 21 13 27 30 9 24 1 20 6 31 16 15 19 22 17 12 5 18 2 10 4 3 7 29 28 8 14 12 | 12 6 4 3 31 2 26 19 21 29 30 1 14 13 22 27 17 24 8 28 9 15 25 18 23 7 16 20 10 11 5 13 | 13 20 24 10 8 12 22 5 17 4 14 6 1 11 27 16 9 26 21 2 19 7 28 3 29 18 15 23 25 31 30 14 | 14 7 25 17 20 30 2 26 29 10 27 15 16 1 12 13 4 28 23 5 24 31 19 21 3 8 11 18 9 6 22 15 | 15 21 5 24 3 16 14 12 25 19 11 8 31 7 1 6 28 30 10 27 2 23 22 4 9 29 20 17 26 18 13 16 | 16 8 19 4 18 27 30 2 10 9 13 31 11 15 14 1 25 5 3 22 28 20 26 29 17 23 6 21 24 7 12 17 | 17 26 15 13 5 21 18 20 14 16 29 24 4 9 3 10 1 7 30 8 6 28 31 12 27 2 25 22 11 19 23 18 | 18 9 14 22 25 7 6 11 2 30 8 17 21 3 20 23 12 1 28 15 13 4 16 26 5 24 29 19 27 10 31 19 | 19 27 18 31 14 9 10 29 6 7 24 22 26 5 25 28 20 3 1 17 23 12 21 11 15 13 2 16 8 30 4 20 | 20 10 12 5 4 6 11 16 26 2 7 3 18 23 31 8 22 13 24 1 27 17 14 19 28 9 21 25 30 29 15 21 | 21 24 16 12 19 8 7 6 30 27 23 4 29 17 18 3 14 15 5 31 1 25 11 2 22 28 10 26 13 9 20 22 | 22 11 17 23 15 26 19 25 18 21 2 13 12 27 5 30 3 9 7 24 10 1 4 20 8 6 14 31 29 16 28 23 | 23 25 13 30 24 20 31 15 22 12 18 10 3 29 8 21 27 11 26 6 16 9 1 5 2 19 17 28 14 4 7 24 | 24 12 8 6 27 4 17 3 15 31 25 2 28 26 9 19 7 21 16 29 18 30 23 1 11 14 5 13 20 22 10 25 | 25 30 20 15 12 10 29 21 11 6 9 5 19 28 4 24 31 23 13 3 8 22 18 16 1 27 26 14 7 2 17 26 | 26 13 21 20 16 24 9 10 7 8 28 12 2 22 19 5 18 17 15 4 3 14 29 6 31 1 30 11 23 27 25 27 | 27 31 9 29 7 22 5 28 3 17 12 11 13 16 30 14 10 19 18 26 25 6 24 23 21 20 1 8 4 15 2 28 | 28 14 23 7 13 25 4 17 31 20 19 30 5 2 24 26 8 29 11 10 21 27 3 15 6 16 22 1 18 12 9 29 | 29 28 11 14 26 23 8 7 27 13 3 25 10 4 21 17 16 31 22 20 15 19 6 30 12 5 9 2 1 24 18 30 | 30 15 10 21 6 5 28 24 23 3 22 16 27 14 2 12 29 25 20 19 4 11 9 8 18 31 13 7 17 1 26 31 | 31 29 22 28 17 11 16 14 19 26 6 23 20 8 15 7 5 27 9 13 30 3 12 25 24 10 18 4 2 21 1
With GF(\(3^2\)):
x + y | 0 1 2 3 4 5 6 7 8 ------|--------------------------- 0 | 0 1 2 3 4 5 6 7 8 1 | 1 2 0 4 5 3 7 8 6 2 | 2 0 1 5 3 4 8 6 7 3 | 3 4 5 6 7 8 0 1 2 4 | 4 5 3 7 8 6 1 2 0 5 | 5 3 4 8 6 7 2 0 1 6 | 6 7 8 0 1 2 3 4 5 7 | 7 8 6 1 2 0 4 5 3 8 | 8 6 7 2 0 1 5 3 4 x * y | 0 1 2 3 4 5 6 7 8 ------|--------------------------- 0 | 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 2 | 0 2 1 6 8 7 3 5 4 3 | 0 3 6 4 7 1 8 2 5 4 | 0 4 8 7 2 3 5 6 1 5 | 0 5 7 1 3 8 2 4 6 6 | 0 6 3 8 5 2 4 1 7 7 | 0 7 5 2 6 4 1 8 3 8 | 0 8 4 5 1 6 7 3 2 x / y | 1 2 3 4 5 6 7 8 ------|------------------------ 0 | 0 0 0 0 0 0 0 0 1 | 1 2 5 8 3 7 6 4 2 | 2 1 7 4 6 5 3 8 3 | 3 6 1 5 4 2 8 7 4 | 4 8 3 1 7 6 5 2 5 | 5 7 8 6 1 4 2 3 6 | 6 3 2 7 8 1 4 5 7 | 7 5 4 3 2 8 1 6 8 | 8 4 6 2 5 3 7 1
With GF(\(3^3\)):
== G(3^3) == x + y | 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 ------|------------------------------------------------------------------------------------------------------------ 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 1 | 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 2 | 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 3 | 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 4 | 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 5 | 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 6 | 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 7 | 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 8 | 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 9 | 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 10 | 10 11 9 13 14 12 16 17 15 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 11 | 11 9 10 14 12 13 17 15 16 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 12 | 12 13 14 15 16 17 9 10 11 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 13 | 13 14 12 16 17 15 10 11 9 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 14 | 14 12 13 17 15 16 11 9 10 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 15 | 15 16 17 9 10 11 12 13 14 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 16 | 16 17 15 10 11 9 13 14 12 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 17 | 17 15 16 11 9 10 14 12 13 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 18 | 18 19 20 21 22 23 24 25 26 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 | 19 20 18 22 23 21 25 26 24 1 2 0 4 5 3 7 8 6 10 11 9 13 14 12 16 17 15 20 | 20 18 19 23 21 22 26 24 25 2 0 1 5 3 4 8 6 7 11 9 10 14 12 13 17 15 16 21 | 21 22 23 24 25 26 18 19 20 3 4 5 6 7 8 0 1 2 12 13 14 15 16 17 9 10 11 22 | 22 23 21 25 26 24 19 20 18 4 5 3 7 8 6 1 2 0 13 14 12 16 17 15 10 11 9 23 | 23 21 22 26 24 25 20 18 19 5 3 4 8 6 7 2 0 1 14 12 13 17 15 16 11 9 10 24 | 24 25 26 18 19 20 21 22 23 6 7 8 0 1 2 3 4 5 15 16 17 9 10 11 12 13 14 25 | 25 26 24 19 20 18 22 23 21 7 8 6 1 2 0 4 5 3 16 17 15 10 11 9 13 14 12 26 | 26 24 25 20 18 19 23 21 22 8 6 7 2 0 1 5 3 4 17 15 16 11 9 10 14 12 13 x * y | 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 ------|------------------------------------------------------------------------------------------------------------ 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 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 2 | 0 2 1 6 8 7 3 5 4 18 20 19 24 26 25 21 23 22 9 11 10 15 17 16 12 14 13 3 | 0 3 6 9 12 15 18 21 24 5 8 2 14 17 11 23 26 20 7 1 4 16 10 13 25 19 22 4 | 0 4 8 12 16 11 24 19 23 14 15 10 26 18 22 2 3 7 25 20 21 1 5 6 13 17 9 5 | 0 5 7 15 11 13 21 26 19 23 25 18 2 4 6 17 10 12 16 9 14 22 24 20 1 3 8 6 | 0 6 3 18 24 21 9 15 12 7 4 1 25 22 19 16 13 10 5 2 8 23 20 26 14 11 17 7 | 0 7 5 21 19 26 15 13 11 16 14 9 1 8 3 22 20 24 23 18 25 17 12 10 2 6 4 8 | 0 8 4 24 23 19 12 11 16 25 21 20 13 9 17 1 6 5 14 10 15 2 7 3 26 22 18 9 | 0 9 18 5 14 23 7 16 25 15 24 6 11 20 2 13 22 4 21 3 12 26 8 17 19 1 10 10 | 0 10 20 8 15 25 4 14 21 24 7 17 23 3 13 19 2 9 12 22 5 11 18 1 16 26 6 11 | 0 11 19 2 10 18 1 9 20 6 17 25 8 16 24 7 15 26 3 14 22 5 13 21 4 12 23 12 | 0 12 24 14 26 2 25 1 13 11 23 8 22 7 10 6 9 21 19 4 16 3 15 18 17 20 5 13 | 0 13 26 17 18 4 22 8 9 20 3 16 7 11 21 12 25 2 10 23 6 24 1 14 5 15 19 14 | 0 14 25 11 22 6 19 3 17 2 13 24 10 21 8 18 5 16 1 12 26 9 23 7 20 4 15 15 | 0 15 21 23 2 17 16 22 1 13 19 7 6 12 18 20 8 14 26 5 11 10 25 4 3 9 24 16 | 0 16 23 26 3 10 13 20 6 22 2 15 9 25 5 8 12 19 17 21 1 4 11 24 18 7 14 17 | 0 17 22 20 7 12 10 24 5 4 9 26 21 2 16 14 19 6 8 13 18 25 3 11 15 23 1 18 | 0 18 9 7 25 16 5 23 14 21 12 3 19 10 1 26 17 8 15 6 24 13 4 22 11 2 20 19 | 0 19 11 1 20 9 2 18 10 3 22 14 4 23 12 5 21 13 6 25 17 7 26 15 8 24 16 20 | 0 20 10 4 21 14 8 25 15 12 5 22 16 6 26 11 1 18 24 17 7 19 9 2 23 13 3 21 | 0 21 15 16 1 22 23 17 2 26 11 5 3 24 9 10 4 25 13 7 19 20 14 8 6 18 12 22 | 0 22 17 10 5 24 20 12 7 8 18 13 15 1 23 25 11 3 4 26 9 14 6 19 21 16 2 23 | 0 23 16 13 6 20 26 10 3 17 1 21 18 14 7 4 24 11 22 15 2 8 19 12 9 5 25 24 | 0 24 12 25 13 1 14 2 26 19 16 4 17 5 20 3 18 15 11 8 23 6 21 9 22 10 7 25 | 0 25 14 19 17 3 11 6 22 1 26 12 20 15 4 9 7 23 2 24 13 18 16 5 10 8 21 26 | 0 26 13 22 9 8 17 4 18 10 6 23 5 19 15 24 14 1 20 16 3 12 2 25 7 21 11 x / y | 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 ------|-------------------------------------------------------------------------------------------------------- 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 | 1 2 19 21 24 11 12 15 25 23 6 7 22 18 8 20 26 14 3 16 4 13 10 5 9 17 2 | 2 1 11 15 12 19 24 21 14 16 3 5 17 9 4 10 13 25 6 23 8 26 20 7 18 22 3 | 3 6 1 16 25 2 14 23 19 13 18 21 10 7 24 4 22 11 9 26 12 17 8 15 5 20 4 | 4 8 20 1 13 10 26 2 17 6 24 19 5 25 23 21 9 22 12 3 16 18 15 11 14 7 5 | 5 7 9 22 1 18 2 17 3 20 21 26 24 16 19 14 8 6 15 10 11 4 25 13 23 12 6 | 6 3 2 23 14 1 25 16 11 26 9 15 20 5 12 8 17 19 18 13 24 22 4 21 7 10 7 | 7 5 18 17 2 9 1 22 6 10 15 13 12 23 11 25 4 3 21 20 19 8 14 26 16 24 8 | 8 4 10 2 26 20 13 1 22 3 12 11 7 14 16 15 18 17 24 6 23 9 21 19 25 5 9 | 9 18 3 26 19 6 11 13 1 17 7 16 8 21 25 12 10 2 5 22 14 20 24 23 15 4 10 | 10 20 22 11 16 17 23 19 26 1 4 14 18 12 21 5 6 13 8 2 15 3 7 25 24 9 11 | 11 19 14 5 4 25 8 7 12 21 1 9 13 3 20 22 23 24 2 15 10 16 17 18 6 26 12 | 12 24 4 3 17 8 22 6 20 18 25 1 15 19 13 16 5 10 14 9 26 7 23 2 11 21 13 | 13 26 23 24 5 16 7 12 15 14 22 8 1 10 9 6 19 21 17 25 18 11 3 4 20 2 14 | 14 25 12 9 20 24 10 18 4 7 19 3 23 1 17 26 15 8 11 5 22 21 13 6 2 16 15 | 15 21 5 10 3 7 6 20 9 4 16 22 25 26 1 11 24 18 23 8 2 12 19 17 13 14 16 | 16 23 21 4 18 15 9 8 7 24 13 20 11 17 6 1 14 5 26 12 3 25 2 10 22 19 17 | 17 22 13 25 15 26 21 14 23 11 10 24 3 8 5 18 1 16 20 19 7 2 9 12 4 6 18 | 18 9 6 13 11 3 19 26 2 22 5 23 4 15 14 24 20 1 7 17 25 10 12 16 21 8 19 | 19 11 25 7 8 14 4 5 24 15 2 18 26 6 10 17 16 12 1 21 20 23 22 9 3 13 20 | 20 10 17 19 23 22 16 11 13 2 8 25 9 24 15 7 3 26 4 1 21 6 5 14 12 18 21 | 21 15 7 20 6 5 3 10 18 8 23 17 14 13 2 19 12 9 16 4 1 24 11 22 26 25 22 | 22 17 26 14 21 13 15 25 16 19 20 12 6 4 7 9 2 23 10 11 5 1 18 24 8 3 23 | 23 16 15 8 9 21 18 4 5 12 26 10 19 22 3 2 25 7 13 24 6 14 1 20 17 11 24 | 24 12 8 6 22 4 17 3 10 9 14 2 21 11 26 23 7 20 25 18 13 5 16 1 19 15 25 | 25 14 24 18 10 12 20 9 8 5 11 6 16 2 22 13 21 4 19 7 17 15 26 3 1 23 26 | 26 13 16 12 7 23 5 24 21 25 17 4 2 20 18 3 11 15 22 14 9 19 6 8 10 1