IBM completes successful field trials on Fully Homomorphic Encryption
The comment by "Ibmresearcher" says that:<p>> Once one can do multiplication and addition all other operations can be bootstrapped from there so you can indeed do anything a Turing complete computer can do.<p>This is apparently iterated on many articles about FHE, but it seems false to me: to do Turing-complete stuff you also need comparisons, and the ability to change your flow depending on comparisons. But clearly you cannot do that with FHE, otherwise you could extract the encrypted content, one bit at a time.<p>My understanding is that an FHE computer can only execute a fixed net of additions and multiplications (or whatever operations they've got). So you can emulate an "if" clause by computing both branches and then selecting one of the two multiplying by 0 and 1 appropriately; you can do bounded "for" loops always executing the maximal number of times, but you can't do unbounded loops, therefore bye bye Turing completeness.<p>Of course there are a lot of interesting algorithms that can be executed without being Turing complete, but the general statement is false. Am I missing anything?