Again, from here:

1. How to convert from polynomial to normal bases?

For example we have polynomial $p(x)=x^{3}+x^{2}+1$
The PB representation in $GF\left (2^{3} \right )$ is
If $GF\left (p^{m} \right )$ be a field with $p^{m}$ elements and $\beta$ an element of it such that $m$ elements $\left \{\beta ,\beta ^{p}, ... , \beta ^{p^{m-1}} \right \}$ are linearly dependent. Then this set forms a normal basis for $GF\left (p^{m} \right )$
The NB representation of elements in $GF\left (2^{3} \right )$ will only use 3 elemens $\beta$, $\beta ^{2}$ dan $\beta ^{4}$.