ECE4253 Digital Communications Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

Addition and Multiplication Tables in Galois Fields GF(2m)

This page presents addition and multiplication tables for Galois fields GF(2m).

Using the Galois Field GF(23) = GF(8) based on the primitive P(x) = x3 + x + 1 = (1011) = 11 (decimal)

Values in GF(23) are 3-bits each, spanning the decimal range [0..7]. Addition takes place on these 3-bit binary values using bitwise XOR.

For example: 6 + 4 = (110) + (100) = (010) = 2   (highlighted below)

The choice of polynomial P(x) plays no role in the addition operation.

 + 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

Multiplication Table

Values in GF(23) are 3-bits each, spanning the decimal range [0..7]. Multiplication takes place on 3-bit binary values (with modulo 2 addition) and then the result is computed modulo P(x) = (1011) = 11 (decimal).

For example: 4 × 3 = (100) × (011) =  (1100)  = (111)  mod (1011) = 7   (highlighted below)

The specific polynomial P(x) provides the modulus for the multiplication results.

 × 1 2 3 4 5 6 7 1 1 2 3 4 5 6 7 2 2 4 6 3 1 7 5 3 3 6 5 7 4 1 2 4 4 3 7 6 2 5 1 5 5 1 4 2 7 3 6 6 6 7 1 5 3 2 4 7 7 5 2 1 6 4 3

Select a primitive polynomial P(x)

 2024-06-17 14:18:30 ADT Last Updated: 2011-02-02 Richard Tervo [ tervo@unb.ca ]