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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.

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Although current FHE schemes are based on tricking SHE schemes , it does not necessarily mean that , that is the only way . great , would be waiting for that reference , as its needed for my research , thanks in advance for your time ! – sashank Jul 20 '12 at 1:19
by chance did you get the paper that made the distinction ? – sashank Jul 21 '12 at 2:50
The intro to this paper goes into the problem. They talk about the tricks necessary to get "pure" FHE. I thought there was something more explicit, but I can't seem to find it. – mikeazo♦ Jul 21 '12 at 11:23

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