EE4253 Digital Communications Department of Electrical and Computer Engineering - University of New Brunswick, Fredericton, NB, Canada

Addition and Multiplication Tables in 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: 4 + 6 = (100) + (110) = (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 × 5 = (100) × (101) =  (10100)  = (010)  mod (1011) = 2   (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)

 Sun May 19 20:24:11 ADT 2013 Last Updated: 02 FEB 2011 Richard Tervo [ tervo@unb.ca ] Back to the course homepage...