Fully homomorphic encryption has been implemented by Craig Gentry using ideal lattices. Can it be applied to data stored on a private cloud setup in practice? I've seen references here and on other sites that lead to the conclusion that it is very slow.
There is no proof that FHE cannot be implemented efficiently. It is just that, right now, we do not know how to do it. With currently known algorithms (Gentry's algorithm, with a few known optimizations), it would be extremely slow, which means impractical (and quite far in the "impractical" realm, actually). FHE would have to become thousands of times faster than what can be done today, in order to achieve some kind of generic practicality.
Of course, there are a few specific applications which can already make use, in a very practical way, of homomorphic encryption. Typically voting systems, such as Helios Voting -- these do not need fully homomorphic encryption, and can work with a plain, efficient, partially homomorphic ElGamal.
The answer to this question is already completely covered by these questions:
Excerpts: "The short answer is that none of them are practical ... yet. But there is a lot of active research, and if we're lucky, maybe that will lead to enough improvements that it might become practical. We'll see." " They're too slow for most/all practical applications. No point in considering homomorphic encryption for production use today -- way too slow."
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