Understanding Cryptography by Christof Paar and Jan Pelzl - Chapter 4 Solutions - Ex4.6

- 2 mins

Exercise 4.6

Compute in $GF(2^8)$:

where the irreducible polynomial is the one used by AES, $P(x) = x^8 + x^4 + x^3 + x + 1$.

Note that Table 4.2 contains a list of all multiplicative inverses for this field.

Solution

I haven’t yet verified this solution independently. If you spot any mistakes, please leave a comment in the Disqus box at the bottom of the page.

The multiplicative inverse could be found via the Euclidian Algorithm, though in this instance I have simply looked it up in the table mentioned above:

Next we perform a naive multiplication of $(x^4 + x + 1)$ with the inverse we just looked up:

All that’s left to do now is to reduce it via the reduction polynomial

The result of this calculation is therefore $x^7 + x^2 + x$. This answer can be verified as correct (assuming my code is correct) by placing the following python code in the __main__ section of the script written for Ex4.3: