= Solution
Since $\alpha$ generates $K^\times$, it cannot lie in a proper subfield, so its <minimal polynomial> over $\mathbb F_2$ has degree $d$. Write it as
$$
P(X)=X^d+c_{d-1}X^{d-1}+\cdots+c_1X+c_0.
$$
The identity $P(\alpha)=0$ and characteristic two give
$$
\alpha^d=\sum_{j=0}^{d-1}c_j\alpha^j.
$$
Multiplying by $\alpha^n$ and applying the linear map $T$ yields
$$
x_{n+d}=T(\alpha^{n+d})
=\sum_{j=0}^{d-1}c_jT(\alpha^{n+j})
=\sum_{j=0}^{d-1}c_jx_{n+j}.
$$
This is a binary linear recurrence of order at most $d$, so $(x_n)$ is the output of a <linear-feedback shift register> of length at most $d$.
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