[mti]

There is an even more fundamental reason why FHE cannot realistically be used for arbitrary computation: it is that some computations have much larger asymptomatic complexity on encrypted data compared to plaintext.

A critical example is database search: searching through a database on n elements is normally done in O(log n), but it becomes O(n) when the search key is encrypted. This means that fully homomorphic Google search is fundamentally impractical, although the same cannot be said of fully homomorphic DNN inference.

[blintz]

There has been a theoretical breakthrough that makes search a O(log n) problem, actually, (https://eprint.iacr.org/2022/1703) but it is pretty impractical (and not getting much faster).

[mti]

Good point. Note however that PIR is a rather restricted form of search (e.g., with no privacy for the server), but even so, DEPIR has polylog(n) queries (not log n), and requires superlinear preprocessing and a polynomial blowup in the size of the database. I think recent concrete estimates are around a petabyte of storage for a database of 2^20 words. So as you say, pretty impractical.