In practice there aren’t any “untrusted collaborative networks” except cryptocurrencies like bitcoin. You either fully trust all the nodes in the network (becau

In practice there aren’t any “untrusted collaborative networks” except cryptocurrencies like bitcoin. You either fully trust all the nodes in the network (because you own them all), as in large distributed systems used by Google, Facebook, etc, etc, or you have some way of _verifying_ submitted results. Verification needs to be cheap, easy, and accurate. In Bitcoin every node in the network verifies that each transaction obeys the rules of the ledger (the transaction signatures are cryptographically verified, but also it checks for double–spending and so on). Any transaction that fails verification is simply dropped. Since verification is far easier than mining a new block, this keeps malicious blocks out of the ledger. Homeomorphic encryption is the middle ground. Solutions still need to be verified, but because only you can understand the encrypted tasks, it is in principle impossible for the other participants to cheat. In practice I think it is still entirely an academic exercise; I don’t think anyone actually uses homeomorphic encryption for anything real.