Communication Complexity of Equality comparison (Catrina and de Hoogh)
What does an invocation mean?


The communication cost of opening 1 secret-shared field element.

Since the protocols of Catrina and de Hoogh are mostly independent of the underlying MPC protocol, this cost can vary. In SPDZ, for example, the trivial opening algorithm requires $O(n^2)$ field elements to be sent across the network in all, but it can also be done in $O(n)$ (see p25).


  Can Costs #1 and #3 also be moved to a pre-processing phase?


Step 1 just generates random bits, so can be done as preprocessing. Step 3 consists of two parts: the first (line 4 of Protocol 4.2 in the paper) can be preprocessed, while the second (line 9) depends on the inputs, $[a_i]$, so must be done online.


  Is Catrina and de Hoogh the most computationally efficient constant-round protocol currently out there that does a secure equals-zero test of a secret value to generate a secret result?


There's a paper by Lipmaa and Toft with a constant-round equality test with less communication than Catrina and de Hoogh's. However, it has higher local computation costs. I don't know of a computationally more efficient protocol.