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# Zero-knowledge proof

In cryptography, a zero-knowledge proof is an interactive method for one party to prove to another that a (usually mathematical) statement is true, without revealing anything other than the verity of the statement.

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.

A zero-knowledge proof must satisfy three properties:

1. 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.
2. Soundness: if the statement is false, no cheating prover can convince the honest verifier that it is true, except with some small probability.
3. Zero-knowledgeness: 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.

A common use of a zero-knowledge proof is in authentication systems where one party wants to be able 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.

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.

## Example strategy

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. 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 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 the following game:

1. Peggy labels the vertices of G with random numbers. An edge can then be represented as a pair of these numbers. She lists the edges of G, and encrypts each edge with a different encryption key. She then sends the encrypted edges to Victor.
2. Victor flips a coin.
• If the coin comes up heads, Peggy surrenders the encryption keys and the mapping from vertices to random numbers. Victor then decrypts the edges and verifies that the encrypted edges sent in step 1 do in fact make up graph G and not some other graph.
• If the coin comes up tails, Peggy surrenders the encryption keys only for the edges that actually form the Hamiltonian cycle. Victor decrypts these edges and verifies that they do indeed form a cycle of the correct length.

An impostor ('Pamela') can try to impersonate Peggy, and 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.

The other strategy that Pamela can follow is to prepare an encryption of some other graph H 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.

The information revealed by Peggy does not give Victor any information at all about the Hamiltonian cycle of G. To see this, note that Victor can manufacture a transcript of the game without talking to Peggy at all. He can select a sequence of heads and tails, and then prepare hypothetical replies from Peggy, without ever knowing the Hamiltonian cycle himself, by following the appropriate impostor strategy in each round. The transcript itself, and the information it contains, has no clue about the legitimacy of Peggy's identity. Peggy proves her identity not because she can give the right answers, but because she can give the right answers without knowing what the questions will be.