Performing simple arithmetic operations on compressed, and both compressed and encrypted data We all know it is possible to perform addition and multiplication operations on encrypted data by means of homomorphic encryption methods. My questions are:

Is there any cryptosystem that is able to perform these operations on


compressed data
first-encrypted then-compressed data
first-compressed then-encrypted data


without any decryption or decompression?
Using generic homomorphic encryption the answer for all three is essentially yes. Although 3. in general is probably mainly of theoretical interest as it would be impractically slow.


For simply compressed data this is simple since compression should not do anything to hide the data. Just decompress, do the operation and possibly compress the result if needed. 
For encrypted-then-compressed data this is also easy. Simply decompress and do the operation using the homomorphic encryption and compress the resulting ciphertext. However, you should note that ciphertexts rarely compress very well so it is not often you gain anything from doing this.
For compress-then-encrypt things get a little more difficult. However, in principle fully homomorphic encryption allows you to compute any function on a ciphertext. This includes decompression of your compression function. So what you could do is to run the decompression algorithm using the homomorphic encryption to get an encryption of the decompressed message, then apply any operation needed and finally apply the compress function using the homomorphic encryption to get an encryption of the compressed result. The problem here is that fully homomorphic encryption is really not practically efficient for anything but very simple computations. So doing compression and decompression would probably be completely impractical.