Multi-party secure computations are general important procedures to compute any function while keeping the security of private inputs. In this work we ask whether preprocessing can allow low latency (that is, small round) secure multi-party protocols that are universally-composable (UC). In particular, we allow any polynomial time preprocessing as long as it is independent of the exact circuit and actual inputs of the specific instance problem to solve, with only a bound k on the number of gates in the circuits known. To address the question, we first define the model of “Multi-Party Computation on Encrypted Data” (MP-CED), implicitly described in [FH96,JJ00,CDN01,DN03]. In this model, computing parties establish a threshold public key in a preprocessing stage, and only then private data, encrypted under the shared public key, is revealed. The computing parties then get the computational circuit they agree upon and evaluate the circuit on the encrypted data. The MP-CED model is ...