Zero-knowledge proof protocols: theoretical foundations and applications in modern cryptography

The article presents a comprehensive overview of zero-knowledge proof (ZKP) protocols as a fundamental concept of modern cryptography. The historical background of their emergence and the main properties ensuring reliability and confidentiality, i.e., completeness, soundness, and zero-knowledge — are considered. A classification of protocols into interactive and non-interactive ones is provided, with a special focus on modern solutions such as the zk-SNARK and the zk-STARK.

The mathematical foundations of ZKPs are described in detail, including discrete logarithm proofs, the use of homomorphic encryption, polynomial commitments, hashing, and elliptic curves. Practical application areas are analyzed, including cryptocurrencies (Zcash, Ethereum), authentication systems, digital identity, and electronic voting. The advantages of using ZKPs are shown, such as enhanced privacy, reduced need for trusted intermediaries, and strengthened security.

At the same time, key challenges are outlined, including scalability, implementation complexity, the problem of trusted setup, and potential vulnerability to quantum computing. It is concluded that zero-knowledge proof protocols are a powerful tool for ensuring confidentiality and reliability of digital systems, while further research is aimed at creating more efficient and quantum-resistant solutions.