Is there difference between Algebraic Homomorphic Encryption and Fully Homomorphic Encryption Schemes?
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It seems that the answer depends on who you ask. Some would say that they are the same. Personally I feel there is a difference. To me, an algebraically homomorphic cryptosystem is one that supports unlimited multiplications and additions of ciphertexts due only to the mathematical structure. A fully homomorphic cryptosystem is one that supports unlimited multiplications and additions of ciphertexts. Thus all algebraically homomorphic cryptosystems are fully homomorphic, but not the other way around.
Given this distinction, none of the existing fully homomorphic cryptosystems are algebraically homomorphic. All existing systems take a somewhat homomorphic cryptosystem and use some tricks to make them fully homomorphic (most commonly Gentry's bootstrapping idea).
PS. I read a paper recently which made this distinction, but cannot seem to find it. I'll keep looking so you'll have a reference other than me.