I would like to ask how to modify Pohlig Hellman algorithm if I need to work with polynomials $GF(2^{60})$ I know how this algorithm works with numbers, but I am not able to imagine how to do some operations with polynomials. E.g. if I need to do $g^{\frac{p-1}{q^e}}$. How can I do this with polynomial? I have found some similar question, but there is not a lot of information, and I didn't understand it.
$p = x^{60} + x^{59} + 1$,
$g = x$,
$h = x^{58} +x^{55} +x^{54} +x^{52} +x^{51} +x^{50} +x^{49} +x^{48} +x^{44} +x^{43} +x^{41} +x^{40} +x^{37} +x^{34} +x^{33} + x^{32} +x^{29} +x^{27} +x^{26} +x^{21} +x^{19} +x^{17} +x^{15} +x^{13} +x^{10} +x^9 +x^6 +x^5 +x^4 +x^2 +1$
$q = p - 1 = x^{60} + x^{59} = x^{59}*(x+1)$ (I am not sure if it is correct factorization)
$ e = 1 , 1$