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What are the advantages of ZeroDB over a database that uses homomorphic encryption like CryptDB?

Do they cater to different uses?

CryptDB doesn't use homomorfic encryption. It is doing sortable deterministic encryption. E.g. after encryption the server can determine whether a > b or a == b. They have a very good understanding of what data does it leak, when can you do that and when cannot.

In our db, random observer cannot see how data are ordered or whether elements are equal to each other.

Homomorfic encryption would be ideal of course, but it is slow and impractical for the moment

Actually, CryptDB does (or at least did originally) use homomorphic encryption (HE) for evaluating SQL arithmetic expressions. While the earliest suggestions for HE were terribly slow, CryptDB found one that was practical. If memory serves, there have since been HE improvements claimed by IBM and/or Google.

But as michwill states (and CryptDB acknowledges), their approach either leaks information or precludes classes of expressions. For example, their order-preserving encryption could be susceptible to a brute-force attack via 'a > b' ... '(a+1) > b' ... '(a+2) > b'

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