View source for Non-interactive zero-knowledge proof

'''Non-interactive [[Zero-knowledge proof|zero-knowledge proof]]s''' are cryptographic primitives, where information between a prover and a verifier can be authenticated by the prover, without revealing any of the specific information beyond the validity of the statement itself. This function of encryption makes direct communication between the prover and verifier unnecessary, effectively removing any intermediaries. The core trustless cryptography "proofing" involves a [[Hash function|hash function]] generation of a random number, constrained within mathematical parameters (primarily to modulate hashing difficulties) determined by the prover and verifier.<ref>{{cite journal |last1=Goldreich |first1=Oded |last2=Krawczyk |first2=Hugo |date=1996 |title=On the Composition of Zero-Knowledge Proof Systems |url=https://epubs.siam.org/doi/abs/10.1137/S0097539791220688 |journal=SAIM |volume=25 |issue=1 |pages=169–192 |doi=10.1137/S0097539791220688 |access-date=4 November 2022}}</ref>
The key advantage of non-interactive [[Zero-knowledge proof|zero-knowledge proof]]s is that they can be used in situations where there is no possibility of interaction between the prover and verifier, such as in online transactions where the two parties are not able to communicate in real time. This makes non-interactive zero-knowledge proofs particularly useful in decentralized systems like blockchains, where transactions are verified by a network of nodes and there is no central authority to oversee the verification process.<ref name=":0">{{Cite book |last1=Gong |first1=Yinjie |last2=Jin |first2=Yifei |last3=Li |first3=Yuchan |last4=Liu |first4=Ziyi |last5=Zhu |first5=Zhiyi |title=2022 International Conference on Big Data, Information and Computer Network (BDICN) |chapter=Analysis and comparison of the main zero-knowledge proof scheme |date=January 2022 |chapter-url=https://ieeexplore.ieee.org/document/9758531 |pages=366–372 |doi=10.1109/BDICN55575.2022.00074|isbn=978-1-6654-8476-3 |s2cid=248267862 }}</ref>

Most non-interactive zero-knowledge proofs are based on mathematical constructs like elliptic curve cryptography or pairing-based cryptography, which allow for the creation of short and easily verifiable proofs of the truth of a statement. Unlike interactive zero-knowledge proofs, which require multiple rounds of interaction between the prover and verifier, non-interactive zero-knowledge proofs are designed to be efficient and can be used to verify a large number of statements simultaneously.<ref name=":0" />

== History ==<!-- really, as of Oct 2020, this is just the academic history of zero-knowledge proofs; needs expansion to include the history of use in real applications of software and apps -->
[[Biography:Manuel Blum|Blum]], Feldman, and [[Biography:Silvio Micali|Micali]]<ref name="bfm">Manuel Blum, Paul Feldman, and Silvio Micali. Non-Interactive Zero-Knowledge and Its Applications. Proceedings of the twentieth annual ACM symposium on Theory of computing (STOC 1988). 103–112. 1988</ref> 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. [[Biography:Oded Goldreich|Goldreich]] and Oren<ref name=goldreich1994>Oded Goldreich and Yair Oren. Definitions and Properties of Zero-Knowledge Proof Systems. Journal of Cryptology. Vol 7(1). 1–32. 1994 [http://www.wisdom.weizmann.ac.il/~oded/PS/oren.ps (PS)]</ref> gave impossibility results{{clarify|what are "impossibility results"|date=October 2020}} for one shot zero-knowledge protocols in the [[Standard model (cryptography)|standard model]]. In 2003, [[Biography:Shafi Goldwasser|Shafi Goldwasser]] and [[Biography:Yael Tauman Kalai|Yael Tauman Kalai]] published an instance of an identification scheme for which any hash function will yield an insecure digital signature scheme.<ref name=goldwasser2003>Shafi Goldwasser and Yael Kalai. On the (In)security of the Fiat–Shamir Paradigm. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS'03). 2003</ref> These results are not contradictory, as the impossibility result{{clarify|date=October 2020}} of Goldreich and Oren does not hold in the [[Common reference string model|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.{{citation needed|date=October 2020}}

The model influences the properties that can be obtained from a zero-knowledge protocol. Pass<ref>Rafael Pass. On Deniability in the Common Reference String and Random Oracle Model. Advances in Cryptology – CRYPTO 2003. 316–337. 2003 [http://www.nada.kth.se/~rafael/papers/denzk.ps (PS)]</ref> 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]].

=== Blockchain applications ===
[[File:STARK proofs diagram.jpg|400px|thumb|A comparison of the most widely used proof systems]]
In 2012, Alessandro Chiesa et al developed the zk-SNARK protocol, an acronym for ''[[Zero-knowledge proof|zero-knowledge]] succinct non-interactive [[Proof of knowledge|argument of knowledge]]''.<ref name=bitansky2012>{{cite book |last1=Bitansky|first1=Nir |last2=Canetti|first2=Ran |last3=Chiesa|first3=Alessandro |last4=Tromer|first4=Eran |title=Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12 |chapter=From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again |chapter-url=http://dl.acm.org/citation.cfm?id=2090263 |publisher=[[Organization:Association for Computing Machinery|ACM]] |doi=10.1145/2090236.2090263 |date=January 2012 |pages=326–349 |isbn=978-1-4503-1115-1 |s2cid=2576177 }}</ref> The first widespread application of zk-SNARKs was in the [[Zcash|Zerocash]] [[Finance:Blockchain|blockchain]] protocol, where zero-knowledge cryptography provides the computational backbone, by facilitating mathematical proofs that one party has possession of certain information without revealing what that information is.<ref name=sasson2016>{{cite web|last1=Ben-Sasson|first1=Eli |last2=Chiesa|first2=Alessandro |last3=Garman|first3=Christina |last4=Green|first4=Matthew |last5=Miers|first5=Ian |last6=Tromer|first6=Eran |last7=Virza|first7=Madars |title=Zerocash: Decentralized Anonymous Payments from Bitcoin |url=http://zerocash-project.org/media/pdf/zerocash-extended-20140518.pdf |publisher=IEEE |access-date=26 January 2016 |date=18 May 2014 }}</ref> Zcash utilized zk-SNARKs to facilitate four distinct transaction types: private, shielding, deshielding, and public. This protocol allowed users to determine how much data was shared with the public ledger for each transaction.<ref>{{cite web |last1=Ben-Sasson|first1=Eli |last2=Chiesa|first2=Alessandro |title=What are zk-SNARKs? |url=https://z.cash/technology/zksnarks/ |publisher=z.cash |access-date=3 November 2022}}</ref> [[Finance:Ethereum|Ethereum]] zk-Rollups also utilize zk-SNARKs to increase scalability.<ref>{{Cite web |title=Zero-Knowledge rollups |url=https://ethereum.org/ |access-date=2023-02-25 |website=ethereum.org |language=en}}</ref>

In 2017, ''Bulletproofs''<ref>{{Cite book |last1=Bünz |first1=Benedikt |last2=Bootle |first2=Jonathan |last3=Boneh |first3=Dan |last4=Poelstra |first4=Andrew |last5=Wuille |first5=Pieter |last6=Maxwell |first6=Greg |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=Bulletproofs: Short Proofs for Confidential Transactions and More |date=May 2018 |chapter-url=https://ieeexplore.ieee.org/document/8418611 |pages=315–334 |doi=10.1109/SP.2018.00020|isbn=978-1-5386-4353-2 |s2cid=3337741 }}</ref> 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.<ref>{{cite book |last1=Bünz |first1=Benedikt |last2=Bootle |first2=Jonathan |last3=Boneh |first3=Dan |last4=Poelstra |first4=Andrew |last5=Wuille |first5=Pieter |last6=Maxwell |first6=Greg |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=Bulletproofs: Short Proofs for Confidential Transactions and More |date=May 2018 |pages=315–334 |doi=10.1109/SP.2018.00020 |isbn=978-1-5386-4353-2 |s2cid=3337741 |chapter-url=https://web.stanford.edu/~buenz/pubs/bulletproofs.pdf |access-date=2 December 2022}}</ref> Bulletproofs was later implemented into Mimblewimble protocol (the basis for Grin and Beam, and [[Finance:Litecoin|Litecoin]] via extension blocks) and [[Finance:Monero (cryptocurrency)|Monero cryptocurrency]].<ref>{{cite web |last1=Odendaal |first1=Hansie |last2=Sharrock |first2=Cayle |last3=Heerden |first3=SW |title=Bulletproofs and Mimblewimble |url=https://tlu.tarilabs.com/cryptography/bulletproofs-and-mimblewimble/MainReport.html#current-and-past-efforts |publisher=Tari Labs University |access-date=3 December 2020 |archive-url=https://web.archive.org/web/20200929160834/https://tlu.tarilabs.com/cryptography/bulletproofs-and-mimblewimble/MainReport.html |archive-date=29 September 2020}}</ref>

In 2018, the ''zk-STARK''  ([[Zero-knowledge proof|zero-knowledge]] Scalable Transparent [[Proof of knowledge|Argument of Knowledge]])<ref>[http://www.cs.technion.ac.il/RESEARCH_DAY_17/POSTERS/michael_riabzev.pdf http://www.cs.technion.ac.il/RESEARCH_DAY_17/POSTERS/michael_riabzev.pdf]</ref> protocol was introduced by Eli Ben-Sasson, Iddo Bentov, Yinon Horesh, and Michael Riabzev,<ref name="iacr2018">{{cite web |author=Eli Ben-Sasson |author2=Iddo Bentov |author3=Yinon Horesh |author4=Michael Riabzev |date=March 6, 2018 |title=Scalable, transparent, and post-quantum secure computational integrity |url=https://eprint.iacr.org/2018/046.pdf |access-date=October 24, 2021 |publisher=[[Organization:International Association for Cryptologic Research|International Association for Cryptologic Research]]}}</ref> offering transparency (no trusted setup), quasi-linear proving time, and poly-logarithmic verification time.
''Zero-Knowledge Succinct Transparent Arguments of Knowledge'' are a type of cryptographic proof system that enables one party (the prover) to prove to another party (the verifier) that a certain statement is true, without revealing any additional information beyond the truth of the statement itself. zk-STARKs are succinct, meaning that they allow for the creation of short proofs that are easy to verify, and they are transparent, meaning that anyone can verify the proof without needing any secret information.<ref name="iacr2018" />

Unlike the first generation of zk-SNARKs, zk-STARKs, by default, do not require a trusted setup, which makes them particularly useful for decentralized applications like blockchains. Additionally, zk-STARKs can be used to verify many statements at once, making them scalable and efficient.<ref name=":0" />

In 2019, HALO recursive zk-SNARKs without a trusted setup were presented.<ref name=":1" /> Pickles<ref>{{Cite web |title=Meet Pickles SNARK: Enabling Smart Contracts on Coda Protocol |url=https://minaprotocol.com/blog/meet-pickles-snark-enabling-smart-contracts-on-coda-protocol |access-date=2023-02-25 |website=Mina Protocol}}</ref> zk-SNARKs, based on the former construction, power MINA, the lightest blockchain.<ref>{{Cite web |last1=Bonneau |first1=Joseph |last2=Meckler |first2=Izaak |last3=Rao |first3=V. |last4=Evan |last5=Shapiro |date=2021 |title=Mina: Decentralized Cryptocurrency at Scale |s2cid=226280610 |language=en}}</ref>

A list of zero-knowledge proof protocols and libraries is provided below along with comparisons based on transparency, universality, and plausible post-quantum security. A transparent protocol is one that does not require any trusted setup and uses public randomness. A universal protocol is one that does not require a separate trusted setup for each circuit. Finally, a plausibly post-quantum protocol is one that is not susceptible to known attacks involving quantum algorithms.
{| class="wikitable"
|+ Non-interactive zero-knowledge proof systems
! ZKP system
! Publication year
! Protocol
! Transparent
! Universal
! Plausibly post-quantum secure
|-
|Pinocchio<ref>{{Cite book |last1=Parno |first1=Bryan |last2=Howell |first2=Jon |last3=Gentry |first3=Craig |last4=Raykova |first4=Mariana |title=2013 IEEE Symposium on Security and Privacy |chapter=Pinocchio: Nearly Practical Verifiable Computation |date=May 2013 |chapter-url=https://ieeexplore.ieee.org/document/6547113 |pages=238–252 |doi=10.1109/SP.2013.47|isbn=978-0-7695-4977-4 |s2cid=1155080 }}</ref>
|2013
|zk-SNARK
|No
|No
|No
|-
|Geppetto<ref>{{Cite book |last1=Costello |first1=Craig |last2=Fournet |first2=Cédric |last3=Howell |first3=Jon |last4=Kohlweiss |first4=Markulf |last5=Kreuter |first5=Benjamin |last6=Naehrig |first6=Michael |last7=Parno |first7=Bryan |last8=Zahur |first8=Samee |title=2015 IEEE Symposium on Security and Privacy |chapter=Geppetto: Versatile Verifiable Computation |date=May 2015 |chapter-url=https://ieeexplore.ieee.org/document/7163030 |pages=253–270 |doi=10.1109/SP.2015.23|isbn=978-1-4673-6949-7 |s2cid=3343426 }}</ref>
|2015
|zk-SNARK
|No
|No
|No
|-
|2013
|zk-SNARK
|No
|No
|No
|-
|Buffet<ref>{{Cite book |title=Efficient RAM and Control Flow in Verifiable Outsourced Computation |url=https://www.ndss-symposium.org/ndss2015/ndss-2015-programme/efficient-ram-and-control-flow-verifiable-outsourced-computation/ |access-date=2023-02-25 |website=NDSS Symposium |year=2015 |language=en-US |doi=10.14722/ndss.2015.23097 |last1=Wahby |first1=Riad S. |last2=Setty |first2=Srinath |last3=Ren |first3=Zuocheng |last4=Blumberg |first4=Andrew J. |last5=Walfish |first5=Michael |isbn=978-1-891562-38-9 }}</ref>
|2015
|zk-SNARK
|No
|No
|No
|-
|ZoKrates<ref>{{Cite book |last1=Eberhardt |first1=Jacob |last2=Tai |first2=Stefan |title=2018 IEEE International Conference on Internet of Things (IThings) and IEEE Green Computing and Communications (GreenCom) and IEEE Cyber, Physical and Social Computing (CPSCom) and IEEE Smart Data (SmartData) |chapter=ZoKrates - Scalable Privacy-Preserving Off-Chain Computations |date=July 2018 |chapter-url=https://ieeexplore.ieee.org/document/8726497 |location=Halifax, NS, Canada |publisher=IEEE |pages=1084–1091 |doi=10.1109/Cybermatics_2018.2018.00199 |isbn=978-1-5386-7975-3|s2cid=49473237 }}</ref>
|2018
|zk-SNARK
|No
|No
|No
|-
|xJsnark<ref>{{Cite book |last1=Kosba |first1=Ahmed |last2=Papamanthou |first2=Charalampos |last3=Shi |first3=Elaine |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=XJsnark: A Framework for Efficient Verifiable Computation |date=May 2018 |chapter-url=https://ieeexplore.ieee.org/document/8418647 |pages=944–961 |doi=10.1109/SP.2018.00018|isbn=978-1-5386-4353-2 |s2cid=13741899 }}</ref>
|2018
|zk-SNARK
|No
|No
|No
|-
|vRAM<ref>{{Cite book |last1=Zhang |first1=Yupeng |last2=Genkin |first2=Daniel |last3=Katz |first3=Jonathan |last4=Papadopoulos |first4=Dimitrios |last5=Papamanthou |first5=Charalampos |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=VRAM: Faster Verifiable RAM with Program-Independent Preprocessing |date=May 2018 |chapter-url=https://ieeexplore.ieee.org/document/8418645 |pages=908–925 |doi=10.1109/SP.2018.00013|isbn=978-1-5386-4353-2 |s2cid=41548742 }}</ref>
|2018
|zk-SNARG
|No
|Yes
|No
|-
|vnTinyRAM<ref>{{Cite book |last1=Ben-Sasson |first1=Eli |last2=Chiesa |first2=Alessandro |last3=Tromer |first3=Eran |last4=Virza |first4=Madars |date=2014 |title=Succinct {Non-Interactive} Zero Knowledge for a von Neumann Architecture |url=https://www.usenix.org/conference/usenixsecurity14/technical-sessions/presentation/ben-sasson |language=en |pages=781–796 |isbn=978-1-931971-15-7}}</ref>
|2014
|zk-SNARK
|No
|Yes
|No
|-
|MIRAGE<ref>{{Cite journal |last1=Kosba |first1=Ahmed |last2=Papadopoulos |first2=Dimitrios |last3=Papamanthou |first3=Charalampos |last4=Song |first4=Dawn |date=2020 |title=MIRAGE: Succinct Arguments for Randomized Algorithms with Applications to Universal zk-SNARKs |url=https://eprint.iacr.org/2020/278 |journal=Cryptology ePrint Archive |language=en}}</ref>
|2020
|zk-SNARK
|No
|Yes
|No
|-
|Sonic<ref>{{Cite book |last1=Maller |first1=Mary |last2=Bowe |first2=Sean |last3=Kohlweiss |first3=Markulf |last4=Meiklejohn |first4=Sarah |title=Proceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security |chapter=Sonic |date=2019-11-06 |chapter-url=https://doi.org/10.1145/3319535.3339817 |series=CCS '19 |location=New York, NY, USA |publisher=Association for Computing Machinery |pages=2111–2128 |doi=10.1145/3319535.3339817 |isbn=978-1-4503-6747-9|s2cid=60442921 |url=https://www.research.ed.ac.uk/en/publications/739b94f1-54f0-4ec3-9644-3c95eea1e8f5 }}</ref>
|2019
|zk-SNARK
|No
|Yes
|No
|-
|2020
|zk-SNARK
|No
|Yes
|No
|-
|PLONK<ref>{{Cite journal |last1=Gabizon |first1=Ariel |last2=Williamson |first2=Zachary J. |last3=Ciobotaru |first3=Oana |date=2019 |title=PLONK: Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge |url=https://eprint.iacr.org/2019/953 |journal=Cryptology ePrint Archive |language=en}}</ref>
|2019
|zk-SNARK
|No
|Yes
|No
|-
|2020
|zk-SNARK
|Yes
|Yes
|No
|-
|Bulletproofs<ref>{{Cite book |last1=Bünz |first1=Benedikt |last2=Bootle |first2=Jonathan |last3=Boneh |first3=Dan |last4=Poelstra |first4=Andrew |last5=Wuille |first5=Pieter |last6=Maxwell |first6=Greg |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=Bulletproofs: Short Proofs for Confidential Transactions and More |date=May 2018 |chapter-url=https://ieeexplore.ieee.org/document/8418611 |pages=315–334 |doi=10.1109/SP.2018.00020|isbn=978-1-5386-4353-2 |s2cid=3337741 }}</ref>
|2018
|Bulletproofs
|Yes
|Yes
|No
|-
|Hyrax<ref>{{Cite book |last1=Wahby |first1=Riad S. |last2=Tzialla |first2=Ioanna |last3=Shelat |first3=Abhi |last4=Thaler |first4=Justin |last5=Walfish |first5=Michael |title=2018 IEEE Symposium on Security and Privacy (SP) |chapter=Doubly-Efficient zkSNARKs Without Trusted Setup |date=May 2018 |chapter-url=https://ieeexplore.ieee.org/document/8418646 |pages=926–943 |doi=10.1109/SP.2018.00060|isbn=978-1-5386-4353-2 |s2cid=549873 }}</ref>
|2018
|zk-SNARK
|Yes
|Yes
|No
|-
|Halo<ref name=":1">{{Cite journal |last1=Bowe |first1=Sean |last2=Grigg |first2=Jack |last3=Hopwood |first3=Daira |date=2019 |title=Recursive Proof Composition without a Trusted Setup |url=https://eprint.iacr.org/2019/1021 |journal=Cryptology ePrint Archive |language=en}}</ref>
|2019
|zk-SNARK
|Yes
|Yes
|No
|-
|Virgo<ref>{{Cite book |last1=Zhang |first1=Jiaheng |last2=Xie |first2=Tiancheng |last3=Zhang |first3=Yupeng |last4=Song |first4=Dawn |title=2020 IEEE Symposium on Security and Privacy (SP) |chapter=Transparent Polynomial Delegation and Its Applications to Zero Knowledge Proof |date=May 2020 |chapter-url=https://ieeexplore.ieee.org/document/9152704 |pages=859–876 |doi=10.1109/SP40000.2020.00052|isbn=978-1-7281-3497-0 |s2cid=209467198 }}</ref>
|2020
|zk-SNARK
|Yes
|Yes
|Yes
|-
|Ligero<ref>{{Cite book |last1=Ames |first1=Scott |last2=Hazay |first2=Carmit |last3=Ishai |first3=Yuval |last4=Venkitasubramaniam |first4=Muthuramakrishnan |title=Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security |chapter=Ligero |date=2017-10-30 |chapter-url=https://doi.org/10.1145/3133956.3134104 |series=CCS '17 |location=New York, NY, USA |publisher=Association for Computing Machinery |pages=2087–2104 |doi=10.1145/3133956.3134104 |isbn=978-1-4503-4946-8|s2cid=5348527 }}</ref>
|2017
|zk-SNARK
|Yes
|Yes
|Yes
|-
|2019
|zk-SNARK
|Yes
|Yes
|Yes
|-
|2019
|zk-STARK
|Yes
|Yes
|Yes
|-
|Zilch<ref>{{Cite web |last=Computing |first=Trustworthy |date=2021-08-30 |title=Transparent Zero-Knowledge Proofs With Zilch |url=https://trustworthy-computing.medium.com/transparent-zero-knowledge-proofs-with-zilch-2031a63fcef3 |access-date=2023-02-25 |website=Medium |language=en}}</ref><ref name="Mouris 2021 3269–3284">{{Cite journal |last1=Mouris |first1=Dimitris |last2=Tsoutsos |first2=Nektarios Georgios |date=2021 |title=Zilch: A Framework for Deploying Transparent Zero-Knowledge Proofs |url=https://ieeexplore.ieee.org/document/9410618 |journal=IEEE Transactions on Information Forensics and Security |volume=16 |pages=3269–3284 |doi=10.1109/TIFS.2021.3074869 |issn=1556-6021 |s2cid=222069813}}</ref>
|2021
|zk-STARK
|Yes
|Yes
|Yes
|}

==Definition==
Originally,<ref name="bfm"/> 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 [[Biography:Adi Shamir|Shamir]]<ref>Uriel Feige, Dror Lapidot, Adi Shamir: Multiple Non-Interactive Zero-Knowledge Proofs Under General Assumptions. SIAM J. Comput. 29(1): 1–28 (1999)</ref> introduced multi-theorem zero-knowledge proofs as a more versatile notion for non-interactive zero-knowledge proofs.

==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 pairing evaluation in a [[Commitment scheme|commitment]]. Using different commitment schemes, this idea was used to build zero-knowledge proof systems under the [[Sub-group hiding|sub-group hiding]]<ref name=groth2006a>Jens Groth, Rafail Ostrovsky, Amit Sahai: Perfect Non-interactive Zero Knowledge for NP. EUROCRYPT 2006: 339–358</ref> and under the decisional linear assumption.<ref name=groth2006b>Jens Groth, Rafail Ostrovsky, Amit Sahai: Non-interactive Zaps and New Techniques for NIZK. CRYPTO 2006: 97–111</ref> These proof systems prove [[Circuit satisfiability problem|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|sub-group hiding]], decisional linear assumption, and [[XDH assumption|external Diffie–Hellman assumption]] that allow directly proving the pairing product equations that are common in [[Pairing-based cryptography|pairing-based cryptography]] have been proposed.<ref>Jens Groth, Amit Sahai: Efficient Non-interactive Proof Systems for Bilinear Groups. EUROCRYPT 2008: 415–432</ref>

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.<ref>Jens Groth. Short Pairing-Based Non-interactive Zero-Knowledge Arguments. ASIACRYPT 2010: 321–340</ref><ref>Helger Lipmaa. Progression-Free Sets and Sublinear Pairing-Based Non-Interactive Zero-Knowledge Arguments. TCC 2012: 169–189</ref>

==References==
<references/>

[[Category:Theory of cryptography]]

{{Sourceattribution|Non-interactive zero-knowledge proof}}