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In cryptography, a zero-knowledge proof or zero-knowledge protocol is an interactive method for one party to prove to another that a (usually mathematical) statement is true, without revealing anything other than the veracity of the statement. A zero-knowledge proof must satisfy three properties: The first two of these are properties of more general interactive proof systems. The third is what makes the proof zero-knowledge. Research in zero-knowledge proofs has been motivated by authentication systems where one party wants to prove its identity to a second party via some secret information (such as a password) but doesn't want the second party to learn anything about this secret. This is called a "zero-knowledge proof of knowledge". However, a password is typically too small or insufficiently random to be used in many schemes for zero-knowledge proofs of knowledge. A zero-knowledge password proof is a special kind of zero-knowledge proof of knowledge that addresses the limited size of passwords. Zero-knowledge proofs are not proofs in the mathematical sense of the term because there is some small probability, the soundness error, that a cheating prover will be able to convince the verifier of a false statement. In other words, they are probabilistic rather than deterministic. However, there are techniques to decrease the soundness error to negligibly small values. One of the most fascinating uses of zero-knowledge proofs within cryptographic protocols is to enforce honest behavior while maintaining privacy. Roughly, the idea is to force a user to prove, using a zero-knowledge proof, that its behavior is correct according to the protocol. Because of soundness, we know that the user must really act honestly in order to be able to provide a valid proof. Because of zero knowledge, we know that the user does not compromise the privacy of its secrets in the process of providing the proof. This application of zero-knowledge proofs was first used in the ground-breaking paper of Goldreich, Micali, and Wigderson on secure multiparty computation.
Cave story There is a well-known story presenting some of the ideas of zero-knowledge proofs, first published by Jean-Jacques Quisquater et al. in their "How to Explain Zero-Knowledge Protocols to Your Children". It is common practice to label the two parties in a zero-knowledge proof as Peggy (the prover of the statement) and Victor (the verifier of the statement). Sometimes P and V are known instead as Pat and Vanna. In this story, Peggy has uncovered the secret word used to open a magic door in a cave. The cave is shaped like a circle, with the entrance in one side and the magic door blocking the opposite side. Victor says he'll pay her for the secret, but not until he's sure that she really knows it. Peggy says she'll tell him the secret, but not until she receives the money. They devise a scheme by which Peggy can prove that she knows the word without telling it to Victor. First, Victor waits outside the cave as Peggy goes in. We label the left and right paths from the entrance A and B. She randomly takes either path A or B. Then, Victor enters the cave and shouts the name of the path he wants her to use to return, either A or B, chosen at random. Providing she really does know the magic word, this is easy: she opens the door, if necessary, and returns along the desired path. Note that Victor does not know which path she has gone down. However, suppose she does not know the word. Then, she can only return by the named path if Victor gives the name of the same path that she entered by. Since Victor chooses A or B at random, she has at most a 50% chance of guessing correctly. If they repeat this trick many times, say 20 times in a row, her chance of successfully anticipating all of Victor's requests becomes vanishingly small, and Victor is convinced that she knows the secret. You may ask, why not just make Peggy take a known path that will force her through the door, and make Victor wait at the entrance? Certainly, that will prove that Peggy knows the secret word, but it also opens the door for eavesdropping. By randomising the initial path that Peggy takes and preventing Victor from knowing it, it reduces the chances that Victor can follow Peggy and learn not just that she knows the secret word, but what the secret word actually is. This part of the exchange is important for keeping the amount of information revealed to a minimum. Example We can extend these ideas to a more realistic cryptography application. In this scenario, Peggy knows a Hamiltonian cycle for a large graph, G. She will prove that she knows this information without revealing the cycle itself. A Hamiltonian cycle in a graph is just one way to implement a zero knowledge proof, in fact any NP problem can be used, (see * for an example with factoring). However, Peggy does not want to simply reveal the Hamiltonian cycle or any other information to Victor; she wishes to keep the cycle secret (perhaps Victor is interested in buying it but wants verification first, or maybe Peggy is the only one who knows this information and is proving her identity to Victor). To show that Peggy knows this Hamiltonian cycle, she and Victor play several rounds of a game. During each round, Peggy does not know which question she will be asked until after giving Victor H. Therefore, in order to be able to answer both, H must be isomorphic to G and she must have a Hamiltonian cycle in H. Because only someone who knows a Hamiltonian cycle in G would always be able to answer both questions, Victor (after a sufficient number of rounds) becomes convinced that Peggy does know this information. However, Peggy's answers do not reveal the original Hamiltonian cycle in G. Each round, Victor will learn only Hs isomorphism to G or a Hamiltonian cycle in H. He would need both answers for a single H to discover the cycle in G, so the information remains unknown as long as Peggy can generate a unique H every round. Because of the nature of the isomorphic graph and Hamiltonian cycle problems, (namely, that they are NP), Victor gains no information about the Hamiltonian cycle in G from the information revealed in each round. If Peggy does not know the information, she can guess which question Victor will ask and generate either a graph isomorphic to G or a Hamiltonian cycle for an unrelated graph, but since she does not know a Hamiltonian cycle for G she cannot do both. With this guesswork, her chance of fooling Victor is , where is the number of rounds. For all realistic purposes, it is infeasibly difficult to defeat a zero knowledge proof with a reasonable number of rounds in this way. History and results Zero-knowledge proofs were first conceived in 1985 by Shafi Goldwasser, et al., in a draft of "The knowledge complexity of interactive proof-systems" While this landmark paper did not invent interactive proof systems, it did invent the IP hierarchy of interactive proof systems (see interactive proof system) and conceived the concept of knowledge complexity, a measurement of the amount of knowledge about the proof transferred from the prover to the verifier. They also gave the first zero-knowledge proof for a concrete problem, that of deciding quadratic nonresidues mod m. In their own words:
The quadratic nonresidue problem has both an NP and a co-NP algorithm, and so lies in the intersection of NP and co-NP. This was also true of several other problems for which zero-knowledge proofs were subsequently discovered, such as an unpublished proof system by Oded Goldreich verifying that a two-prime modulus is not a Blum integer. Oded Goldreich, et al., took this one step further, showing that, assuming the existence of unbreakable encryption, one can create a zero-knowledge proof system for the NP-complete graph coloring problem with three colors. Since every problem in NP can be efficiently reduced to this problem, this means that, under this assumption, all problems in NP have zero-knowledge proofs. The reason for the assumption is that, as in the above example, their protocols require encryption. A commonly cited sufficient condition for the existence of unbreakable encryption is the existence of one-way functions, but it is conceivable that some physical means might also achieve it. On top of this, they also showed that the graph nonisomorphism problem, the complement of the graph isomorphism problem, has a zero-knowledge proof. This problem is in co-NP, but is not currently known to be in either NP or any practical class. More generally, Goldreich, Goldwasser et al. would go on to show that, also assuming unbreakable encryption, there are zero-knowledge proofs for all problems in IP=PSPACE, or in other words, anything that can be proved by an interactive proof system can be proved with zero knowledge. Not liking to make unnecessary assumptions, many theorists sought a way to eliminate the necessity of one way functions. One way this was done was with multi-prover interactive proof systems (see interactive proof system), which have multiple independent provers instead of only one, allowing the verifier to "cross-examine" the provers in isolation to avoid being misled. It can be shown that, without any intractability assumptions, all languages in NP have zero-knowledge proofs in such a system It turns out that in an Internet-like setting, where multiple protocols may be executed concurrently, building zero-knowledge proofs is more challenging. The line of research investigating concurrent zero-knowledge proofs was initiated by the work of Dwork, Naor, and Sahai. One particular development along these lines has been the development of witness-indistinguishable proof protocols. The property of witness-indistinguishability is related to that of zero-knowledge, yet witness-indistinguishable protocols do not suffer from the same problems of concurrent execution. See also | ||||||||
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