I think you've got the right general idea, but I don't know that it has to be homomorphic encryption. After all, an index of the public web is not really secret, and the user doesn't have a private key for it.
In the simplest case, you could make a search engine in the form of a big, public, regularly-updated database, and let users send in arbitrary queries (run in a sandbox/quota environment).
That's essentially what we've got now, except the query parser is a proprietary black box that changes all the time. I don't see any inherent reason they couldn't expose a lower-level interface, and let browsers build queries. Why can't web browsers be responsible for converting a user's text (or voice) into a search engine query structure?
In the simplest case, you could make a search engine in the form of a big, public, regularly-updated database, and let users send in arbitrary queries (run in a sandbox/quota environment).
That's essentially what we've got now, except the query parser is a proprietary black box that changes all the time. I don't see any inherent reason they couldn't expose a lower-level interface, and let browsers build queries. Why can't web browsers be responsible for converting a user's text (or voice) into a search engine query structure?