Trading help for interaction in statistical zero-knowledge proofs

We define interactive and non-interactive statistical zero-knowledge proofs with (limited) help, as proofs that can be almost perfectly simulated, where the prover and the verifier share a reference string that is computed by a probabilistic polynomial-time trusted third party that receives as input the statement to be proven (i.e. the input to the protocol). We compare these models with the standard interactive and non-interactive SZK models, trying to understand when this form of help can replace the interaction between the prover and the verifier and vice versa. We show that every promise problem that has an SZK protocol with help also has one without help. As for the opposite, we show non-interactive SZK proofs with help for natural languages for which only interactive SZK proofs are known. In order to achieve that, we introduce a complete problem for the class of promise problems that have non-interactive SZK proofs with help.