Zero-knowledge proof

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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:

Completeness: if the statement is true, the honest verifier (that is, one following the protocol properly) will be convinced of this fact by an honest prover.
Soundness: if the statement is false, no cheating prover can convince the honest verifier that it is true, except with some small probability.
Zero-knowledge: if the statement is true, no cheating verifier learns anything other than this fact. This is formalized by showing that every cheating verifier has some simulator that, given only the statement to be proven (and no access to the prover), can produce a transcript that "looks like" an interaction between the honest prover and the cheating verifier.

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 (called the soundness error) that a cheating prover will be able to convince the verifier of a false statement. However, there are standard techniques to decrease the soundness error to any arbitrarily small value.

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.

1 Cave story
2 Example
3 History and results
4 References
5 See also
6 External links

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Cave story

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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".[1] 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 magic door at one end and the entrance at the other. 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.

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.

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Example

We can extend these ideas to a more realistic cryptography scenario. Say that Peggy's public key is a large graph, which we will call G. Peggy generated G at some time in the past, and then published it widely. Because she manufactured it specially for the purpose, Peggy knows a Hamiltonian cycle in G (most easily, she could have created the edges forming the cycle first and then added other edges). Peggy will prove her identity to Victor by proving that she knows a Hamiltonian cycle in G. Even though G is public information, nobody else can do this, because nobody else knows a Hamiltonian cycle of G, and finding Hamiltonian cycles in graphs is believed to be a difficult problem (see NP-completeness).

However, Peggy mustn't simply reveal the Hamiltonian cycle to Victor, since then Victor (or an eavesdropper) would be able to impersonate Peggy in the future. Peggy mustn't reveal any information at all about the cycle, because an eavesdropper might gather information on several different occasions and assemble it into enough information to be able to impersonate Peggy.

To prove her identity, Peggy and Victor play several rounds of a game.

At the beginning of each round, Peggy gives Victor an encrypted graph. She asserts that this graph is an encrypted version of her public key graph, G, and that she knows a Hamiltonian cycle in this graph.
During each round, Victor is allowed to ask Peggy one question. He can either ask her to prove that the encrypted graph she sent is in fact G, or he can ask her to prove that she does in fact know a Hamiltonian cycle in this graph. He decides which question to ask by flipping a coin.
If Peggy is asked to prove that the graph is in fact G, she provides the decryption keys for the graph. Victor can then decrypt the graph and compare it to the published form of G.
If Peggy is asked to prove that she knows a Hamiltonian cycle in the graph, she provides the decryption keys for the edges in the cycle. Victor can then decrypt those edges and verify that they do in fact form a Hamiltonian cycle.

During any given round, Peggy does not know which question she will be asked until after she has already given Victor the graph. Therefore she must be prepared to answer both questions, which means that the graph she gives Victor must be equivalent to G and that she must be able to demonstrate a Hamiltonian cycle in the graph. Because only Peggy would always be able to answer both questions, Victor (after a sufficient number of rounds) becomes convinced that Peggy is who she says she is.

However, Peggy's answers do not compromise her secret knowledge of the Hamiltonian cycle in G. On 50% of the rounds, Victor will learn of a Hamiltonian cycle in an encrypted graph. Because he is only given enough information to decrypt the cycle, and not the entire graph, he still can't locate the cycle in G. On the other 50% of the rounds, Victor will learn no new information, as Peggy will simply prove that the encrypted graph is G, and G is already public.

An impostor ('Pamela') can try to impersonate Peggy. In order to answer both of Victor's questions correctly, Pamela would have to know a Hamiltonian cycle in G. Because she does not, she only has a 50% chance of successfully fooling Victor in any particular round. There are two possible impersonation strategies. Pamela can send an encryption of Peggy's graph G. In this case, she escapes detection if Victor throws heads; she reveals the encryption, and Victor verifies that the graph is indeed G. But if Victor throws tails, Pamela is caught. She is required to reveal the keys of a set of edges that make up a Hamiltonian cycle of G, and she can't do that, because she doesn't know one (and almost certainly cannot make one because of the NP-completeness of the problem and the large graph size).

The other strategy that Pamela can follow is to prepare an encryption of some other graph H with the same number of edges and vertices, for which she does know a Hamiltonian cycle. In this case she is safe if Victor throws tails; she reveals the cycle, and, since Victor never sees the rest of the edges, he never learns that the graph was H and not G. But if Victor throws heads, Pamela is required to reveal the entire graph, and Victor will see that it is not G.

By playing twenty rounds of this game, Victor can reduce the probability of being fooled by Pamela to a mere 2^-20. By playing more rounds, Victor can reduce the probability as far as desired.

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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"<ref> Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof-systems. Proceedings of 17th Symposium on the Theory of Computation, Providence, Rhode Island. 1985. Draft. Extended abstract</ref> 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:

Of particular interest is the case where this additional knowledge is essentially 0 and we show that [it] is possible to interactively prove that a number is quadratic non residue mod m releasing 0 additional knowledge. This is surprising as no efficient algorithm for deciding quadratic residuosity mod m is known when m’s factorization is not given. Moreover, all known NP proofs for this problem exhibit the prime factorization of m. This indicates that adding interaction to the proving process, may decrease the amount of knowledge that must be communicated in order to prove a theorem.

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<ref> Oded Goldreich. A zero-knowledge proof that a two-prime moduli is not a Blum integer. Unpublished manuscript. 1985.</ref>.

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<ref>Oded Goldreich, Silvio Micali, Avi Wigderson. Proofs that yield nothing but their validity. Journal of the ACM, volume 38, issue 3, p.690-728. July 1991.</ref>. 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<ref>Michael Ben-Or, Oded Goldreich, Shafi Goldwasser, Johan Hastad, Joe Kilian, Silvio Micali, and Phillip Rogaway. Everything provable is provable in zero-knowledge. S. Goldwasser, editor. In Advances in Cryptology--CRYPTO '88, volume 403 of Lecture Notes in Computer Science, p.37-56. Springer-Verlag, 1990, 21-25. August 1988.</ref>.

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<ref>M. Ben-or, Shafi Goldwasser, J. Kilian, and A. Wigderson. Multi prover interactive proofs: How to remove intractability assumptions. Proceedings of the 20th ACM Symposium on Theory of Computing, p.113-121. 1988.</ref>

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<ref> Cynthia Dwork, Moni Naor, and Amit Sahai. Concurrent Zero Knowledge. Journal of the ACM (JACM), v.51 n.6, p.851-898, November 2004.</ref>.

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