We show that any private-key encryption scheme that is weakly homomorphic with respect to addition modulo 2, can be transformed into a public-key encryption scheme. The homomorphic feature referred to is a minimalistic one; that is, the length of a homomorphically generated encryption should be independent of the number of ciphertexts from which it was created. We do not require anything else on the distribution of homomorphically generated encryptions (in particular, we do not require them to be distributed like real ciphertexts). Our resulting public-key scheme is homomorphic in the following sense. If i+1 repeated applications of homomorphic operations can be applied to the private-key scheme, then i repeated applications can be applied to the public-key scheme.
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