Minimum resource zero-knowledge proofs (extended abstracts)

What are the resources of a zero-knowledge Proof? Interaction, communication, and envelops. That interaction, that is the number of rounds of a protocol, is a resource is clear. Actually, it is not a very available one: having someone on the line to answer your questions all the time is quite a luxury. Thus, minimizing the number of rounds in zero-knowledge proofs will make these proofs much more attractive from a practical standpoint. That communication, that is the number of bits exchanged in a protocol, is a resource is also immediately clear. Perhaps, what is less clear is why envelopes are a resource. Let us explain why this is the case.Zero-knowledge proofs work by hiding data from a verifier. Only some of this data will be later revealed, at the verifier's request: enough to convince him that the statement at end is true, but not enough to give him any knowledge beyond that. Data can be hidden in two ways: physically - e.g. by putting it into an envelope - or digitally - by encrypting it. But why is it important to minimize the number of envelopes? Physically, because a GOOD envelope is expensive - it actually must be a led box or a safe. Digitally, because minimizing the number of envelopes corresponds to reducing the transmitted bits. In fact, to transmit an encrypted message, one needs to send more bits than in the message itself. For instance, to send an encrypted bit, one needs to send at least 60 bits in some probabilistic encryption scheme. Also, to decrypt each ciphertext, one has to send the decryption key. However, many bits may be encrypted and decrypted with the same overhead of a few bits. Thus if one manages to package the data that should be hidden in as few envelopes as possible, while maintaining zero-knowledge, the protocol will require transmitting much less bits.