# Addition and multiplication table for GF(2²)

I’m reading Joan Daemen and Vincent Rijmen’s book The Design of Rijndael and I’m giving myself a refresher course on group theory.

Key to the encryption standard is the Galois field on 256 elements GF(28). A multiplication table of 256 elements by 256 elements quickly becomes a wall of text, so let’s reason by analogy and look at GF(22).

There are a number of ways to represent elements of the field; we’ll start by representing them as polynomials with degree at most 1, and with integer coefficients modulo 2. There are four such polynomials: {0, 1, x, x + 1}.

Here are the addition and multiplication tables:

 + 0 1 x x + 1 0 0 1 x x + 1 1 1 0 x + 1 x x x x + 1 0 1 x + 1 x + 1 x 1 0

 ⋅ 0 1 x x + 1 0 0 0 0 0 1 0 1 x x + 1 x 0 x x2 mod m x2 + x mod m x + 1 0 x + 1 x2 + x mod m x2 + 1 mod m

Hold on. What’s that funny-looking m?

It’s a “reduction polynomial” which brings the product back down to degree 1 or less. It has to be a polynomial of degree 2. There are four such polynomials: let’s try each and see what we get.

x2
 0 x x 1
x2 + 1
 1 x + 1 x + 1 0
x2 + x
 x 0 0 x + 1
x2 + x + 1
 x + 1 1 1 x

Note that the first three polynomials all factor into products of lower-degree polynomials: x2 = x(x), x2 + 1 = (x + 1)(x + 1), x2 + x = x(x + 1). Only x2 + x + 1 is prime; and this prime reduction polynomial generates a complete multiplication table with no 0s. This is a necessary condition to be a field. Our final tables are:

 + 0 1 x x + 1 0 0 1 x x + 1 1 1 0 x + 1 x x x x + 1 0 1 x + 1 x + 1 x 1 0

 ⋅ 0 1 x x + 1 0 0 0 0 0 1 0 1 x x + 1 x 0 x x + 1 1 x + 1 0 x + 1 1 x

We can also write our elements in binary form: 0 => 00, 1 => 01, x => 10, and x + 1 => 11. In this notation our tables become:

 + 00 01 10 11 00 00 01 10 11 01 01 00 11 10 10 10 11 00 01 11 11 10 01 00

 ⋅ 00 01 10 11 00 00 00 00 00 01 00 01 10 11 10 00 10 11 01 11 00 11 01 10

Rijndael works in GF(28) and uses a reduction polynomial of x8 + x4 + x3 + x + 1. They say this is prime. I sure hope so.