Let $F_{p^n}$, $F_{p^m}$ finite fields, $p$ prime, $m | n$. $m$ divides $n$, so$F_{p^m}$ is a subfield of $F_{p^n}$.
Elements of $F_{p^m}$ can be represented as polynomials over $F_p$ of degree $< m$, with multiplication defined modulo irreducible polynomial $f(x)$ of degree $m$.
Elements of $F_{p^n}$ can be represented as polynomials over $F_p$ of degree $< n$, with multiplication defined modulo irreducible polynomial $g(x)$ of degree $n$.
Let's have $h(x)$ such a polynomial representation of an element in $F_{p^m}$. How can I compute representation $q(x)$ of the same element in the larger field $F_{p^n}$?
The only algorithm I know is finding (randomly) the generating element of the multiplicative group of the larger field (and power it to find the generator of the multiplicative group of the subfield) and then solving the discrete logarithm in the subfield. However, this approach is not very effective: the discrete logarithm is hard, finding generator is also hard... Is there any better algorithm?