Given an elliptic curve over $\mathbb Z_n$ is it $\#P$ hard to compute

1. $\# E(\mathbb Z_n)$?

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

2. $\# E(\mathbb Z_n)\bmod 2$?

3. $a\bmod 2$ where $2^a|\# E(\mathbb Z_n)$ and $2^{a+1}\nmid\# E(\mathbb Z_n)$?

Given an elliptic curve over $\mathbb Z_n$

1. Is it $\#P$ hard to compute $\# E(\mathbb Z_n)$?

2. Is it $PP$-hard to compute $\# E(\mathbb Z_n)\leq\frac n2$?

3. Is it $\oplus P$ hard to compute $\# E(\mathbb Z_n)\bmod 2$?

Given an Elliptic curve over $\mathbb Z_n$ is it $\#P$ hard to compute

1. $\# E(\mathbb Z_n)$?

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

2. $\# E(\mathbb Z_n)\bmod 2$?

3. $a\bmod 2$ where $2^a|\# E(\mathbb Z_n)$ and $2^{a+1}\nmid\# E(\mathbb Z_n)$?

Given an elliptic curve over $\mathbb Z_n$ is it $\#P$ hard to compute

1. $\# E(\mathbb Z_n)$?

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

2. $\# E(\mathbb Z_n)\bmod 2$?

3. $a\bmod 2$ where $2^a|\# E(\mathbb Z_n)$ and $2^{a+1}\nmid\# E(\mathbb Z_n)$?