I'm learning finite fields behind Advance Encryption Standard. As far as I know, the irreducible polynomial used in AES is $x^8+x^4+x^3+x+1$. This is because AES s-box is based on bytes(8bits). Now I am trying to enlarge the s-box to 4bytes(32bits), so I need to find a polynomials of degree 32 over $Z_2[x]$.
So the question is How can i get a irreducible polynomials of degree 32 over $Z_2[x]$?
I've searched for some approaches and not very familiar with trace most mentioned. What materials relating $Z_2[x]$ can I refer to?
Besides, I would appreciate it if there is some constructed such polynomials for me to test! Thanks!
The appropriate Conway polynomial $$ c = x^{32}+x^{15}+x^9+x^7+x^4+x^3+1 $$ would fit the bill (this can be obtained from a computer algebra system).
To verify that this is primitive, hence irreducible, first factor (in case using a computer algebra system) \begin{align} 4\,294\,967\,295 &= 2^{32} - 1 \\&= (2^{16} -1) \cdot (2^{16} + 1) \\&= \dots \\&= (2^{2} - 1) \cdot (2^{2} + 1) \cdot (2^{4} + 1) \cdot (2^{8} + 1) \cdot (2^{16} + 1) \\&= 3 \cdot 5 \cdot 17 \cdot 257 \cdot 65537. \end{align} Then show (using a computer algebra system) that $x$ has indeed order $B = 2^{32} - 1$ modulo $c$, by checking that $x^{B} \equiv 1 \pmod{c}$, whereas $x^{B/p} \not\equiv 1 \pmod{c}$ for all primes $p \in \{ 3, 5, 17, 257, 65537 \}$ dividing $B$.