Degenerate discrete logarithm in binary field Given a field $\mathbb{F}_{2^n}$, are there any choices of primitive element $g$ that make the discrete logarithm easier for that generator? That is, are there any degenerate cases?

For example, if I choose a small generator like $g = x$ in $\mathbb{F}_{2^{256}}$, can I quickly find $n$ such that $g^n = a$ for a given $a$? Or is this equally difficult for any $g$ picked in any $\mathbb{F}_{2^n}$ over any irreducible polynomial?
It is equally difficult (within a factor of 2) for any irreducible polynomial.

Suppose $g$ was your 'cheap' irreducible polynomial, that is, one for which, given $g^n$, you can rederive $n$ quickly.  Then, given an arbitrary pair $h, h^x$, you can quickly find $a, b$, such that $h = g^a$ and $h^x = g^b$, and then immediately deduce that $x = a^{-1}b$