There are ways to text match against encrypted documents, at least approximately, without requiring homomorphic encryption.
One simple scheme would be to encrypt in ECB mode (which has it's own disadvantages, which I'll ignore for now). Then, say that we have a relatively long string to match, we can encrypt that string with the same key (after trimming to the block size) and simply search for it's encrypted form in the document. If you see a match, and the string isn't an exact multiple of the block size, you need to decrypt a portion of the file (which you can easily do in ECB mode) and perform a more-exact match.
I wouldn't recommend this in a real system, but it's one example of how you can keep data encrypted at rest without requiring complete decryption for string matching.
I don't, on the other hand, know of any results in homomorphic encryption that would allow you to do the same thing with that approach.
One simple scheme would be to encrypt in ECB mode (which has it's own disadvantages, which I'll ignore for now). Then, say that we have a relatively long string to match, we can encrypt that string with the same key (after trimming to the block size) and simply search for it's encrypted form in the document. If you see a match, and the string isn't an exact multiple of the block size, you need to decrypt a portion of the file (which you can easily do in ECB mode) and perform a more-exact match.
I wouldn't recommend this in a real system, but it's one example of how you can keep data encrypted at rest without requiring complete decryption for string matching.
I don't, on the other hand, know of any results in homomorphic encryption that would allow you to do the same thing with that approach.