Efficiently getting bits of N! ?
Dick Lipton has a beautiful post from 2009 on the relationship between the factorial function and factoring. There's a lot there that is unrelated to this question, but one salient point is this theorem:


  If $n!$ can be computed by a
  straight-line arithmetic computation
  in $O(\log^{c} n)$ steps, then
  factoring has polynomial size
  circuits.


I suspect that this is evidence that your question, especially within the time bounds you mention, is going to be difficult to answer.