Decrypting partial message.

I'm wondering whether such interesting symmetric-key encryption exists or not (from theoretical perspective):

We encrypt a vector of messages $(m_1,m_2,\cdots,m_\ell)$ using the symmetric key $K$ to obtain a ciphertext $CT$. Then to decrypt the ciphertext, the decryption algorithm, using the same symmetric key $K$, takes as input an identity $i$ and the ciphertext $CT$ and outputs the partial message $m_i$.

If such symmetric key encryption scheme does exist, what assumption can it be based on?

• uhm CTR-mode can do this or do you want more properties? – SEJPM Jun 24 '17 at 15:15
• @SEJPMYes, as I said I want it from theoretical perspective, so I'm wondering whether such scheme exists without the help of CTR-mode and is solely based on an assumption (e.g., one-way function). – CryptoLover Jun 24 '17 at 15:24
• How is CTR-mode not theoretical? – Elias Jun 24 '17 at 20:44
• I think you can do this with something like a punctured PRF: rather than using the full symmetric key $K$ to decrypt, you modify the key using some puncturing function $P$ which takes the key $K$ and the identity $i$ and outputs a key $K_i$ which can only decrypt the $i$th message. More info: eprint.iacr.org/2013/352.pdf – pg1989 Jun 25 '17 at 1:06