Decrypt (homomorphically and symmetrically) encrypted data using (homomorphically) encrypted key? Is it possible to do the following?


Generate symmetric key $K$
Encrypt data $D$ with the symmetric key : $E_k{(D)}$
Encrypt the symmetric key and the encrypted data using homomorphic encryption scheme (e.g. Paillier) with FHE key : $E_{paillier}(E_k{(D))}$ and $E_{paillier}(K)$
Send both $E_{paillier}(E_k{(D))}$ and $E_{paillier}(K)$ to the other third party
The third party (symmetric) decrypt using it and outputs $E_{paillier}(D)$
Yes, this is possible as long as the homomorphic cipher supports all operations necessary to implement the symmetric encryption algorithm. So for a partially homomorphic cipher like Paillier, this will not be possible. With a fully-homomorphic cipher (or somewhat homomorphic cipher configured with sufficient multiplicative depth), this is very feasible, even for standard symmetric algorithms like AES.

For example see Homomorphic Evaluation of the AES Circuit. AES has become a sort of defacto standard benchmark for fully homomorphic ciphers.

See also Ciphers for MPC and FHE, where the researchers look at designing secure symmetric ciphers that are specifically optimized towards homomorphic evaluation.