In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the verifier) that some given statement is true, without conveying to the verifier any information beyond the mere fact of that statement's truth. The intuition behind the nontriviality of zero-knowledge proofs is that it is trivial to prove possession of the relevant information simply by revealing it; the hard part is to prove this possession without revealing this information (or any aspect of it whatsoever). In light of the fact that one should be able to generate a proof of some statement only when in possession of certain secret information connected to the statement, the verifier, even after having become convinced of the statement's truth by means of a zero-knowledge proof, should nonetheless remain unable to prove the statement to further third parties. Zero-knowledge proofs can be interactive, meaning that the prover and verifier exchange messages according to some protocol, or noninteractive, meaning that the verifier is convinced by a single prover message and no other communication is needed. In the standard model, interaction is required, except for trivial proofs of BPP problems. In the common random string and random oracle models, non-interactive zero-knowledge proofs exist. The Fiat–Shamir heuristic can be used to transform certain interactive zero-knowledge proofs into noninteractive ones.
Abstract examples
The red card proof One example of a math-free zero-knowledge proof is if Peggy wants to prove to Victor that she has drawn a red card from a standard deck of 52 playing cards, without revealing which specific red card she holds. Victor observes Peggy draw a card at random from the shuffled deck, but she keeps the card face-down so he cannot see it. To prove her card is red without revealing its identity, Peggy takes the remaining 51 cards from the deck and systematically shows Victor all 26 black cards (the 13 spades and 13 clubs) one by one, placing them face-up on the table. Since a standard deck contains exactly 26 red cards and 26 black cards, and Peggy has demonstrated that all the black cards remain in the deck, Victor can conclude with certainty that Peggy's hidden card must be red. This proof is zero-knowledge because Victor learns only that Peggy's card is red, but gains no information about whether it is a heart or diamond, or which specific red card she holds. The proof would be equally convincing whether Peggy held the Ace of Hearts or the Two of Diamonds. Furthermore, even if the interaction were recorded, the recording would not reveal Peggy's specific card to future observers, maintaining the zero-knowledge property. If Peggy were lying and actually held a black card, she would be unable to produce all 26 black cards from the remaining deck, making deception impossible. This demonstrates the soundness of the proof system. This type of physical zero-knowledge proof using standard playing cards belongs to a broader class of card-based cryptographic protocols that allow participants to perform secure computations using everyday objects.
Where's Waldo Another well-known example of a zero-knowledge proof is the "Where's Waldo" example. In this example, the prover has a page from a Where's Waldo? children's book, which shows many hundreds of cartoon people, where only one of them is the visually distinctive character of Waldo. The prover wants to prove to the verifier that they know where Waldo is on the page, without revealing his location to the verifier. The prover starts by taking a large black board with a small hole in it, the size of Waldo. The board is twice the size of the book in both directions, so the verifier cannot see where on the page the prover is placing it. The prover then places the board over the page so that Waldo is in the hole. The verifier can now look through the hole and see Waldo, but cannot see any other part of the page. Therefore, the prover has proven to the verifier that they know where Waldo is, without revealing any other information about his location. This example is not a perfect zero-knowledge proof, because the prover does reveal some information about Waldo's location, such as his body position. However, it is a decent illustration of the basic concept of a zero-knowledge proof.
The Ali Baba cave
There is a well-known story presenting the fundamental ideas of zero-knowledge proofs, first published in 1990 by Jean-Jacques Quisquater and others in their paper "How to Explain Zero-Knowledge Protocols to Your Children". The two parties in the zero-knowledge proof story are Peggy as the prover of the statement, and Victor, the verifier of the statement. In this story, Peggy has uncovered the secret word used to open a magic door in a cave. The cave is shaped like a ring, with the entrance on one side and the magic door blocking the opposite side. Victor wants to know whether Peggy knows the secret word; but Peggy, being a very private person, does not want to reveal her knowledge (the secret word) to Victor or to reveal the fact of her knowledge to the world in general. They label the paths from the entrance A and B. First, Victor waits outside the cave as Peggy goes in. Peggy takes either path A or B; Victor is not allowed to see which path she takes. 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 did not know the word. Then, she would only be able to return by the named path if Victor were to give the name of the same path by which she had entered. Since Victor would choose A or B at random, she would have a 50% chance of guessing correctly. If they were to repeat this trick many times, say 20 times in a row, her chance of successfully anticipating all of Victor's requests would be reduced to 1 in 220, or 9.54 × 10−7. Thus, if Peggy repeatedly appears at the exit Victor names, then he can conclude that it is extremely probable that Peggy does, in fact, know the secret word.
… excerpt ends here. Continue reading the full article.
