I am just thinking there might be other better ways to preserve user's search privacy: using LLM embeddings (https://en.wikipedia.org/wiki/Word_embedding).
The browser creates embeddings of user query, then send the embeddings to the server.
To complete a search, the server is a machine and it does not really need text to understand what a user want. A series of numbers, like LLM embeddings, are totally fine (actually it might even be better, because embeddings map similar words closely, like Duck and Bird have similar embeddings).
On the privacy side, LLM embeddings are a bunch of numbers. Even the embeddings are associated with a user, other people cannot make meaning out of the embeddings. Therefore the user's privacy is preserved.
What do you think?