Complexity of factorial exponent over composite moduli I know that computing factorial modulo a composite number has no fast algorithm and showing non-polylogarithmic lower bound in BSS model for factorial would separate P from NP in that model.

Given $a\in\Bbb Z/n\Bbb Z$, where $n$ is composite, what is the complexity of calculating $a^{m!}$ in $\Bbb Z/n\Bbb Z$ for any given integer $n&gt;m&gt;0$?
Borwein's method to compute the exponent and the doubling method for multiplying out the $a$'s wouldn't be too terrible, but as @SashoNikolov pointed out it is exponential with regards to the input size. You can store all intermediate results in $O(log(n))$ bits when multiplying out $a$. 

I don't see what this has to do with P vs NP.