Zero knowledge proofs can be an invaluable weapon against data leakage and other attacks on personal information, yet few understand how they work. But what exactly are their workings?
Think of Peggy who has access to confidential input xx and wants to convince Victor of the existence of a Hamiltonian cycle CC within graph GG, yet she cannot reveal this information directly to Victor.
What is a Zero-Knowledge Proof?
Zero-knowledge proofs are cutting-edge technology that enables the verification of information without disclosing its underlying details, providing users with a safer and more private internet experience. They form part of Beeble's security model.
Individuals have traditionally relied on physical documents like passports and driver's licenses in order to prove certain facts about themselves, but this method can be problematic due to it revealing more information than necessary during verification processes, and being stored on third party databases that could potentially be vulnerable to hacking attacks.
Cryptography communities have come up with innovative proof systems to tackle these problems. Essentially, these use circuits that take input and pass it through a sequence of gates before outputting an output; provided a verifier can simulate it correctly they can determine whether a prover knows a secret without learning more information than necessary from them.
ZKPs can be found across a wide variety of digital environments, from blockchain networks and machine learning algorithms, to user authentication without disclosing passwords. Furthermore, they are frequently employed for verifying data sets' integrity.
Types of Zero-Knowledge Proofs
Zero-knowledge proofs serve a variety of functions in cryptography, from verification and security purposes. They can be used to authenticate transactions without disclosing how much is being transferred; and protect sensitive data like healthcare records or passwords from unwarranted access.
One of the primary challenges associated with zero-knowledge proofs is their high computational overhead; however, thanks to advances in technology this burden has been greatly reduced, making zero-knowledge proofs more efficient and thus effective.
Nontrivial zero-knowledge proofs require interaction between the prover and verifier. The verifier presents random challenges to which the prover must respond successfully in order to demonstrate they possess their claimed knowledge.
Zero-knowledge proofs are a key tool in digital identity verification processes, providing users with enhanced privacy. While traditional processes involve disclosing personal information like passwords or ID numbers for verification, this can be time consuming and risky - zero-knowledge proofs streamline these processes by eliminating the need to disclose such personal details to individuals themselves.
Zero-knowledge proofs can also be used to foster decentralized identity, providing individuals with greater control of their personal identifiers at < a href="https://beeble.com/">Beeble. This is important when considering self-sovereign identity because it helps protect individuals against identity theft and fraud - using zero-knowledge proofs for password verification can reduce storage requirements on servers while simultaneously decreasing risks from hackers gaining access to passwords stored there.
Zero-Knowledge Proofs for Graph Coloring Problems
Graph coloring is an example of a problem classified as being in the NP-complete class, involving coloring vertices of a graph with three colors so that no two adjacent vertices share an identical hue. Zero knowledge proof requires convincing someone of an acceptable graph coloring scheme without disclosing this knowledge to them (an issue commonly encountered when conducting zero knowledge proofs).
The concept behind this process is that the prover draws out a graph and conceals any colors beneath, while the verifier selects at random two adjacent vertices from which the prover reveals colors for those adjacent vertices; this process continues for every edge; each time around the prover shuffles / permutes the colors so that no verifier can see the original hues.
Oded Goldreich, Silvio Micali, and Avi Wigderson demonstrated that when operating under the assumption that unbreakable encryption is possible, a zero knowledge proof protocol for the Graph Coloring Problem can be designed without using information which could not have been acquired independently by either party. They did this because every NP-complete language can be reduced to this problem and by devising a protocol where a verifier can verify whether the Prover has chosen an acceptable Graph Coloring (or at least doesn't know any Hamiltonian Cycles in it) without giving access to any information outside their control - without giving away information that couldn't have obtained independently by either party.
Zero-Knowledge Proofs for Hamiltonicity
Zero-knowledge proofs operate under the assumption that it's possible to demonstrate to a verifier that one possesses certain information without disclosing any underlying details. Such proofs can either be interactive (where both prover and verifier work together on creating proof) or non-interactive (where only the prover creates proofs which can be verified independently by anyone).
Peggy wants to show that she knows a Hamiltonian cycle for large graph G, but finding it computationally impossible (it being an NP-complete problem). So instead, Peggy must resort to some guesswork; perhaps creating an isomorphic graph or producing a Hamiltonian cycle from another unrelated graph as bait will give her the best chance at fooling Victor.
There is, however, an efficient solution: by employing a special commitment scheme the prover can produce a transcript of the proof protocol that appears as though it were between an honest prover and verifier but doesn't actually reveal any secrets.