Conversion from/to power form and polynomial basis is a finite field

Question

I'm trying to figure out if there is a way better than $O(m^2)$ of converting between the power form $(\alpha^n)$ to polynomial basis. Right now, I'm just enumerating the entire field up to the $n$-th entry for power form to polynomial basis. To convert from polynomial basis to power form, I'm enumerating the field until I find an entry that matches.