# Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 4 Solutions - Ex4.3

- 7 mins

## Exercise 4.3

Generate the multiplication table for the extension field $GF(2^3)$ for the case that the irreducible polynomial is $P(x) = x^3 + x + 1$. The multiplication table is in this case a $8 \times 8$ table. (Remark: You can do this manually or write a program for it.)

### Solution

This solution is verified as correct by the official Solutions for Odd-Numbered Questions manual.

$\begin{array}{c|c c c c c} \times & 0 & 1 & x & x + 1 & x^2 & x^2 + 1 & x^2 + x & x^2 + x + 1 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & x & x + 1 & x^2 & x^2 + 1 & x^2 + x & x^2 + x + 1 \\ x & 0 & x & x^2 & x^2 + x & x + 1 & 1 & x^2 + x + 1 & x^2 + 1 \\ x + 1 & 0 & x + 1 & x^2 + x & x^2 + 1 & x^2 + x + 1 & x^2 & 1 & x \\ x^2 & 0 & x^2 & x + 1 & x^2 + x + 1 & x^2 + x & x & x^2 + 1 & 1 \\ x^2 + 1 & 0 & x^2 + 1 & 1 & x^2 & x & x^2 + x + 1 & x + 1 & x^2 + x \\ x^2 + x & 0 & x^2 + x & x^2 + x + 1 & 1 & x^2 + 1 & x + 1 & x & x^2 \\ x^2 + x + 1 & 0 & x^2 + x + 1 & x^2 + 1 & x & 1 & x^2 + x & x^2 & x + 1 \end{array}$

I wrote a python script which can calculate multiplication tables for $GF(2^3)$ fields (the class has since been split into a module):