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.

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.

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?

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.