I am trying to solve a cryptography challenge where I am given a finite field $K = \mathbb{F}_{2^{113}}$ and $K' = \mathbb{F}_{(2^{113})^{4}}$ as well as $P, Q = s\cdot P$ ($s$ is the secret I am trying to recover, $P, Q$ points on an elliptic curve). Through a Tate Pairing (already given) I am able to embed elements onto $K'$. So I need to solve the discrete logarithm in $K'$ as follows:
$$a = b^s$$ where $a,b$ are elements in $K'$ (given). The problem is, that K' is constructed as an extension field over K, meaning we have $K' = K[y]_{mod y^4 + y + 1}$. But $K$ itself is already an extension field, $K = \mathbb{Z}_2[x]_{mod x^{113} + x^9 + 1}$. The library I am currently using to calculate the discrete log is extremely slow in $K'$, whereas if we were in $GF(2^{4\cdot113})$ it would be way faster.
Therefore, how can I find an explicit isomorphism from $K'$ to $GF(2^{4\cdot113})$, without having to solve the DLP (i.e. just mapping a generator of $K'$)? Since both fields have the same order, an isomorphism must exist.
Or is there a way to solve this DLP in $K'$ in a faster way? Most of the papers I have read on this topic, seem to always assume you are already in a "single" extension field and hence your operations are quite fast already.