$E_1$ and $E_2$ are IND-CPA secure encryption schemes. $E$ is defined as: $k_1,k_2 \leftarrow K_1 \times K_2$ . $E_{k_1,k_2}(m) \leftarrow E_{1,k_1}(m)||E_{2,k_2}(m)$.
Hope the notations are in an obvious manner. $\leftarrow$ means randomly choosing an element from the corresponding distribution. || stands for concatenation.
My question is whether the encryption scheme $E$ is IND-CPA secure.
Intuitively, this would be secure. If you can not get any information from either $E_1$ or $E_2$, should you get no information given both. (Of course there is a loophole here: $E_1$ or $E_2$ by itself may not leak information, however the combination of them may give some information away.)
When I tried to prove it I can not reduce the break of $E$ to either $E_1$ or $E_2$. Neither can I find a counter example.