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Are there any ways to perform homomorphic encryption of logical OR (or logical AND) without using fully homomorphic encryption?

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One practical (if not theoretically satisfying) way would be to use Pallier, with the mapping "any encrypted nonzero value is an encrypted TRUE; any encrypted zero value is an encrypted FALSE". To encrypt a TRUE value, you pick a random value and encrypt that, and the logical OR operation is the standard Pallier homomorphic addition.

Now, obviously, the OR of two TRUE values has a tiny probability of becoming FALSE; however it is far more likely that your computer is hit by a meteor...

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