You can actually do it both ways - where the ciphertext is an obfuscated program, and where keys a programs and ciphertexts are short. https://eprint.iacr.org/

You can actually do it both ways - where the ciphertext is an obfuscated program, and where keys a programs and ciphertexts are short. https://eprint.iacr.org/2013/454.pdf https://eprint.iacr.org/2013/454.pdf and friends have a direct description of the latter and references to the former. This is just one mechanism.