Given $N$ and $M$, is it possible to get the $M$'th bit (or digit of any small base) of $N!$ in time/space of $O( p( ln(N), ln(M) ) )$, where $p(x, y)$ is some polynomial function in $x$ and $y$?

i.e. Given $N = 2^\eta$, $M = 2^\mu$ (with $N$, $M \in \mathbb{Z}$), find bit $2^\mu$ of $(2^\eta)!$ in $O( p(\eta, \mu) )$.

Or perhaps I should be asking whether this problem is NP-Complete?

NOTE: Crossposted to cstheory.stackexchange where Suresh Venkat was kind enough to point me to Dick Lipton's post from 2009 that seems to indicate that this problem is as hard as factoring.