# 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?

• How are $\Pi_0$ and $\Pi_1$ combined? ​ ​
– user991
Jul 27, 2016 at 9:09
• function composition Jul 27, 2016 at 9:19
• In which order are they composed and which one is CCA? Jul 27, 2016 at 10:37
• @Maeher this reads like a homework (to me at least) and the twist probably is that you don't know which one has which property and should prove that the resulting scheme is (not) IND-CCA security (depending on the order?). Jul 27, 2016 at 10:58

You should really specify how they are 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 in 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.