How to solve raising a polynomial to the power of a number mod something

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It doesn't help, I want to be able to keep the exponent modulo such that it reduces when I multiply it with its inverse, as in Diffie Helman

The only way I know in which $x^{a \text{ mod } n}$ makes sense is if $x$ has multiplicative order dividing $n$ (so that $x^n = 1$ and hence $x^{a + kn} = x^{a}x^{kn} = x^{a}(x^n)^k = x^{a}\cdot 1 = x^a$, meaning that the "mod" notation in the exponent is well-defined), so in that case you don't have to worry about "keeping the exponent modulo $n$". If $x \in GF(2^{48})$ then exponents modulo $2^{48} - 1$ are well-defined in this way, so there is no harm in lifting to integers (and in Sage it is necessary, to avoid type errors).

It is not this: (x^n)^k that I am afraid of. I want a and b to be of the following form: a*b = 1 mod 2^96-1 such that when i do (x^a)^b, it will give me x. But if I do the lift on b, then it wont maintain the property that it is the inverse of a in mod 2^96-1

What you want to do in your latest comment makes mathematical sense only if the multiplicative order of x divides $2^{96} - 1$ (so that there is no harm in lifting). If this is not the case then you are doing something very strange, redefining exponentiation in an inconsistent way. More likely you are misunderstanding whatever source material you are trying to implement.

Thanks for letting me know, solved it :) made my understand that I should look a bit more at my logic. I will mark this as the solution, however, the inverse was not the issue here.