Degenerate discrete logarithm in binary field
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 $x = a^{-1}b$