The original CPA game:
- The adversary gives two arbitrary messages $m_0$ and $m_1$ to a challenger;
- The challenger encrypts an arbitrary message $m_u$, $u∈\{0,1\}$ , and sends the ciphertext to the adversary;
- The adversary outputs the guess $u$.
Now I give a revised version of the above CPA game:
- The adversary gives two arbitrary sets of messages $M_0$ and $M_1$ to a challenger, where $|M_0|=|M_1|$, and each pair of counterparts are of equal size;
- The challenger encrypts an arbitrary set of message $M_u$, $u∈\{0,1\}$, and sends the ciphertexts to the adversary;
- The adversary outputs the guess $u$.
If a public-key encryption scheme is CPA-secure is it also secure in this revised version of CPA game?