IND-CCA Security of a encryption scheme which is combination of two schemes I have two public key encryption scheme $\Pi_0$ and $\Pi_1$ and one of them is IND-CCA secure, another one is IND-CPA secure. Let $\Pi$ be a public key encryption scheme, which is a combination of $\Pi_0$ and $\Pi_1$. As IND-CCA security implies IND-CPA security, so $\Pi$ is IND-CPA secure.  Is $\Pi$ is IND-CCA secure? How to prove or disprove it?
You should really specify how are they combined. If it is just composition, you should analyze the cases $S_1=\Pi_0\circ \Pi_1$ $S_2=\Pi_1\circ \Pi_0$, $S_3=\Pi_0\circ \Pi_0$ and $S_4=\Pi_1\circ \Pi_1$, where $\Pi_0$ is CCA and $\Pi_1$ is CPA. If you manage to prove a security type all four cases, you may establish rules for a general composition. However, you may also consider the meet in the middle case when composing if that is your definition of IND-CCA.