Given an elliptic curve over $\mathbb Z_n$ is it $\#P$ hard to compute
- $\# E(\mathbb Z_n)$?
I doubt the problem is $PP$-hard since it seems unlikely $\# E(\mathbb Z_n)\leq\frac n2$ is exceeded.
Nevertheless is it $\oplus P$ hard to compute
$\# E(\mathbb Z_n)\bmod 2$?
$a\bmod 2$ where $2^a|\# E(\mathbb Z_n)$ and $2^{a+1}\nmid\# E(\mathbb Z_n)$?
