Can we construct fully homomorphic encryption scheme based on non-circuit approach? At present, all FHE scheme are be constructed based on circuit approach. Can we construct fully homomorphic encryption scheme based on non-circuit approach?  Is Polly cracker non-circuit approach ?
Is Polly cracker non-circuit approach?


Polly Cracker is non-circuit.


  Can we construct fully homomorphic encryption scheme based on non-circuit approach?


The best info for this I could find comes from a recent paper proposing a non-circuit approach due to Xiao, et al.. 

They identify a number of non-circuit systems, but go on to say that "they all either have been broken or lack conclusive security evidence."

They then propose a symmetric system which reduces to the problem of factoring.

So it seems that the answer is yes, one can be constructed. It is still an active area of research, however, and I personally wouldn't use the proposed systems yet.

Now, what I am exactly not quite sure of is what non-circuit means? In the paper I reference, they state: 


  The main works on homomorphic encryption algorithms are circuit based. They develop the encryption schemes and computation algorithms considering a single bit plaintext. All operations on various operand types can then be achieved by constructing the corresponding circuits.


So is a circuit approach simply that the cryto algorithms assume a single bit plaintext? If that is the case, then there are many, many more FHE systems which are "non-circuit". In fact, the BGV system does not require single bit plaintexts. So perhaps I am missing something in the circuit vs non-circuit discussion.