Answer for #1

Again, from here:

1. How to convert from polynomial to normal bases?

Answer:
GF element can be represented in Polynomial Basis (PB) or Normal Basis (NB).

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} .

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