I would like to compare ciphertext based on the order assumed on the message space.
So, do you know if there exists some encryption scheme with the follow property:
Let $c_0 = E(m_0)$ and $c_1 = E(m_1)$ for all $m_0$ and $m_1$ in the message space. So,
$$c_0 \leq_C c_1 \Leftrightarrow m_0 \leq m_1$$
Note that I used $\leq_C$ instead of $\leq$, because the partial order relation on the ciphertext space may be different from the partial order of the message space.
For example, if I have $c_0$ and $c_1$ such that $c_0 = E(-3)$ and $c_1 = E(103)$, I would like to know that $c_0 \leq_C c_1$ without knowing the clear texts $-3$ and $103$.
If something is unclear, just let me know.
Thank you.
