I want to know how to compute this, All this polynomials are over the finite field $\mathbb{F}_{2^8}$.
I have that
$$ (x^6 + x^4 + x^3)^{-1} \equiv x^4 + x^3 \mod(x^8 + x^4 + x^3 + x + 1) $$
Where $m(x) = x^8 + x^4 + x^3 + x + 1$ is the Rijndael polynomial. But I don't know how to compute this.
Hint: Use the extended Euclidean algorithm for $\gcd(x^6 + x^4 + x^3,x^8 + x^4 + x^3 + x + 1)$ and express $1=(x^6 + x^4 + x^3)p(x)+(x^8 + x^4 + x^3 + x + 1)q(x)$. Then you can interpret $p(x)$ as the desired inverse.