Irreducible Polynomial for $GF(2^{256})$

Question

I'm looking for a pattern to generate Galois Field multiplication for $2^{256}$ binary value. So far I have come up with a patter as follows;

$$ 1 \rightarrow 1 \\ x \rightarrow x \\ x^2\rightarrow x^2\\ ...\\ x^{256} \rightarrow x + 1 $$

Is it $x + 1$ for $x^{256}$? If so, for $x^{257} \rightarrow (x^2 + x)$?

Thanks in advance!

Answer 1

See Miodrag Zivkovic, A Table of Primitive Binary Polynomials. The example $x^{256}+x^{241}+x^{178}+x^{121}+1$ is given for the field of size $2^{256}$.