Non-interactive zero-knowledge proof
This section needs expansion with: history of how zero-knowledge proofs are used in real applications and apps, and for what purposes. You can help by adding to it. (October 2020)
Blum, Feldman, and Micali showed in 1988 that a common reference string shared between the prover and the verifier is sufficient to achieve computational zero-knowledge without requiring interaction. Goldreich and Oren gave impossibility results[clarification needed] for one shot zero-knowledge protocols in the standard model. In 2003, Shafi Goldwasser and Yael Tauman Kalai published an instance of an identification scheme for which any hash function will yield an insecure digital signature scheme. These results are not contradictory, as the impossibility result[clarification needed] of Goldreich and Oren does not hold in the common reference string model or the random oracle model. Non-interactive zero-knowledge proofs however show a separation between the cryptographic tasks that can be achieved in the standard model and those that can be achieved in 'more powerful' extended models.
The model influences the properties that can be obtained from a zero-knowledge protocol. Pass showed that in the common reference string model non-interactive zero-knowledge protocols do not preserve all of the properties of interactive zero-knowledge protocols; e.g., they do not preserve deniability.
Non-interactive zero-knowledge proofs can also be obtained in the random oracle model using the Fiat–Shamir heuristic. A 2012 article by Bitansky et al introduced the acronym zk-SNARK for zero-knowledge succinct non-interactive argument of knowledge. The first widespread application of zk-SNARKs was in the Zerocash blockchain protocol, where zero-knowledge crytography provides the computational backbone, by facilitating mathematical proofs that one party has possession of certain information without revealing what that information is. By 2021, "82 cryptocurrencies worth a total of US$8.85 billion [encrypted] their transactions with zero-knowledge proofs or similar private technology."
In 2017, Bulletproofs was released, which enable proving that a committed value is in a range using a logarithmic (in the bit length of the range) number of field and group elements. Bulletproofs was later implemented into Mimblewimble protocol (where Grin and Beam cryptocurrencies based on) and Monero cryptocurrency.
In 2018, the zk-STARK (zero-knowledge Scalable Transparent ARgument of Knowledge) protocol was introduced, offering transparency (no trusted setup), quasi-linear proving time, and poly-logarithmic verification time.
Originally, non-interactive zero-knowledge was only defined as a single theorem proof system. In such a system each proof requires its own fresh common reference string. A common reference string in general is not a random string. It may, for instance, consist of randomly chosen group elements that all protocol parties use. Although the group elements are random, the reference string is not as it contains a certain structure (e.g., group elements) that is distinguishable from randomness. Subsequently, Feige, Lapidot, and Shamir introduced multi-theorem zero-knowledge proofs as a more versatile notion for non-interactive zero knowledge proofs.
In this model the prover and the verifier are in possession of a reference string sampled from a distribution, D, by a trusted setup . To prove statement with witness w, the prover runs and sends the proof, , to the verifier. The verifier accepts if , and rejects otherwise. To account for the fact that may influence the statements that are being proven, the witness relation can be generalized to parameterized by .
Verification succeeds for all and every .
More formally, for all k, all , and all :
Soundness requires that no prover can make the verifier accept a wrong statement except with some small probability. The upper bound of this probability is referred to as the soundness error of a proof system.
More formally, for every malicious prover, , there exists a negligible function, , such that
The above definition requires the soundness error to be negligible in the security parameter, k. By increasing k the soundness error can be made arbitrary small. If the soundness error is 0 for all k, we speak of perfect soundness.
A non-interactive proof system is multi-theorem zero-knowledge, if there exists a simulator, , such that for all non-uniform polynomial time adversaries, ,
Here outputs for and both oracles output failure otherwise.
Pairing-based non-interactive proofs
Pairing-based cryptography has led to several cryptographic advancements. One of these advancements is more powerful and more efficient non-interactive zero-knowledge proofs. The seminal idea was to hide the values for the evaluation of the pairing in a commitment. Using different commitment schemes, this idea was used to build zero-knowledge proof systems under the sub-group hiding and under the decisional linear assumption. These proof systems prove circuit satisfiability, and thus by the Cook–Levin theorem allow proving membership for every language in NP. The size of the common reference string and the proofs is relatively small; however, transforming a statement into a boolean circuit incurs considerable overhead.
Proof systems under the sub-group hiding, decisional linear assumption, and external Diffie–Hellman assumption that allow directly proving the pairing product equations that are common in pairing-based cryptography have been proposed.
Under strong knowledge assumptions, it is known how to create sublinear-length computationally sound proof systems for NP-complete languages. More precisely, the proof in such proof systems consists only of a small number of bilinear group elements.
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- Bitansky, Nir; Canetti, Ran; Chiesa, Alessandro; Tromer, Eran (January 2012). "From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again". Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12. ACM. pp. 326–349. doi:10.1145/2090236.2090263. ISBN 9781450311151. S2CID 2576177.
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