The historical goal of cryptography is to securely transmit or store a message in an insecure medium. In that era, before public key cryptography, we had two kinds of people: those who had the correct key, and those who did not. Nowadays however, we live in a complex world with equally complex goals and requirements: securely passing a note from Alice to Bob is not enough. We want Alice to use her smartphone to vote for Carol, without Bob the tallier, or anyone else learning her vote; we also want guarantees that Alice’s ballot contains a single, valid vote and we want guarantees that Bob will tally the ballots properly. This is in fact made possible because of zero knowledge protocols. This thesis presents research performed in the area of zero knowledge protocols across the following threads: we relax the assumptions necessary for the Damgard, Fazio and ˚ Nicolosi (DFN) transformation, a technique which enables one to collapse a number of three round protocols into a single message. This approach is motivated by showing how it could be used as part of a voting scheme. Then we move onto a protocol that lets us prove that a given computation (modeled as an arithmetic circuit) was performed correctly. It improves upon the state of the art in the area by significantly reducing the communication cost. A second strand of research concerns multi-user signatures, which enable a signer to sign with respect to a set of users. We give new definitions for important primitives in the area as well as efficient instantiations using zero knowledge protocols. Finally, we present two possible answers to the question posed by voting receipts. One is to maximise privacy by building a voting system that provides receipt-freeness automatically. The other is to use them to enable conventual and privacy preserving vote copying
