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'''Secure two-party computation''' (2PC, or '''secure function evaluation''') is a sub-problem of [[secure multi-party computation]] (MPC) that has received special attention by researchers because of its close relation to many [[cryptographic]] tasks.<ref>{{Citation |last1=Wang |first1=Xiao |title=Faster Secure Two-Party Computation in the Single-Execution Setting |date=2017 |url=http://link.springer.com/10.1007/978-3-319-56617-7_14 |work=Advances in Cryptology – EUROCRYPT 2017 |volume=10212 |pages=399–424 |editor-last=Coron |editor-first=Jean-Sébastien |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-56617-7_14 |isbn=978-3-319-56616-0 |access-date=2022-10-19 |last2=Malozemoff |first2=Alex J. |last3=Katz |first3=Jonathan |series=Lecture Notes in Computer Science |editor2-last=Nielsen |editor2-first=Jesper Buus|url-access=subscription }}</ref><ref>{{Cite web |title=MPC Wallet - What is MPC? |url=https://zengo.com/mpc-wallet/ |access-date=2022-10-19 |website=ZenGo |language=en-US}}</ref> The goal of 2PC is to create a generic protocol that allows two parties to jointly compute an arbitrary function on their inputs without sharing the value of their inputs with the opposing party.<ref>{{Cite book |last1=Henecka |first1=Wilko |last2=K ögl |first2=Stefan |last3=Sadeghi |first3=Ahmad-Reza |last4=Schneider |first4=Thomas |last5=Wehrenberg |first5=Immo |title=Proceedings of the 17th ACM conference on Computer and communications security |chapter=TASTY |date=2010 |chapter-url=http://portal.acm.org/citation.cfm?doid=1866307.1866358 |language=en |___location=Chicago, Illinois, US |publisher=ACM Press |pages=451–462 |doi=10.1145/1866307.1866358 |isbn=978-1-4503-0245-6|s2cid=7276194 |url=https://encrypto.de/papers/HKSSW10.pdf }}</ref> One of the most well known examples of 2PC is [[Yao's Millionaires' Problem|Yao's millionaireMillionaires' problem]], in which two parties, Alice and Bob, are millionaires who wish to determine who is wealthier without revealing their wealth.<ref>{{Citation |last1=Lin |first1=Hsiao-Ying |title=An Efficient Solution to the Millionaires' Problem Based on Homomorphic Encryption |date=2005 |work=Applied Cryptography and Network Security |volume=3531 |pages=456–466 |editor-last=Ioannidis |editor-first=John |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/11496137_31 |isbn=978-3-540-26223-7 |last2=Tzeng |first2=Wen-Guey |editor2-last=Keromytis |editor2-first=Angelos |editor3-last=Yung |editor3-first=Moti|doi-access=free }}</ref> Formally, Alice has wealth <math>a</math>, Bob has wealth <math>b</math>, and they wish to compute <math>a \geq b</math> without revealing the values <math>a</math> or <math>b</math>.
[[Andrew Yao|Yao]]'s [[garbled circuit protocol]] for two-party computation only provided security against passive adversaries.<ref>{{Cite book | last1 = Yao | first1 = A. C. | title = 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982) |year=1982 doi|pages=160–164 |chapter=Protocols for secure computations |doi=10.1109/SFCS.1982.38 |s2cid=206558698}}</ref> pagesOne of the first general solutions for achieving security against active adversary was introduced by Goldreich, Micali and Wigderson<ref>{{Cite book|last1=Goldreich|first1=O.|last2=Micali|first2=S.|last3=Wigderson|first3=A.|title=Proceedings 160–164of the nineteenth annual ACM conference on Theory of computing - STOC '87 |chapter=How yearto play ANY mental game |date=1987-01-01|chapter-url=https://doi.org/10.1145/28395.28420|___location=New 1982York, New York, US|publisher=Association pmidfor Computing Machinery|pages=218–229|doi=10.1145/28395.28420|isbn=978-0-89791-221-1|s2cid=6669082 }}</ref> by applying Zero-Knowledge Proof to enforce semi-honest behavior.<ref>{{Cite book |last1=Goldwasser pmc|first1=S |last2=Micali |first2=S |last3=Rackoff |first3=C |title=Proceedings of the seventeenth annual ACM symposium on Theory of computing - STOC '85 |chapter=The knowledge complexity of interactive proof-systems |date=1985-12-01 Protocols|chapter-url=https://doi.org/10.1145/22145.22178 for|___location=Providence, secureRhode computationsIsland, US |publisher=Association s2cidfor Computing Machinery |pages=291–304 206558698|doi=10.1145/22145.22178 |isbn=978-0-89791-151-1|s2cid=8689051 }}</ref> onlyThis providedapproach securitywas againstknown passiveto adversariesbe impractical for years due to high complexity overheads. However, significant improvements have been made toward applying this method in 2PC and Abascal, Faghihi Sereshgi, Hazay, Yuval Ishai and Venkitasubramaniam gave the first efficient protocol based on this approach.<ref>{{Cite book|last1=Abascal|first1=Jackson|last2=Faghihi Sereshgi|first2=Mohammad Hossein|last3=Hazay|first3=Carmit|last4=Ishai|first4=Yuval|last5=Venkitasubramaniam|first5=Muthuramakrishnan|title=Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security |chapter=Is the Classical GMW Paradigm Practical? The Case of Non-Interactive Actively Secure 2PC |date=2020-10-30|chapter-url=https://doi.org/10.1145/3372297.3423366|series=CCS '20|___location=Virtual Event, US|publisher=Association for Computing Machinery|pages=1591–1605|doi=10.1145/3372297.3423366|isbn=978-1-4503-7089-9|s2cid=226228208 }}</ref> Another type of 2PC protocols that are secure against active adversaries were proposed by Yehuda Lindell and Benny Pinkas,<ref>{{Cite book | last1 = Lindell | first1 = Y. | title = Advances in Cryptology - EUROCRYPT 2007 | last2 = Pinkas | first2 = B. | chapter = An Efficient Protocol for Secure Two-Party Computation in the Presence of Malicious Adversaries | doi = 10.1007/978-3-540-72540-4_4 | volume = 4515 | pages = 52–78 | year = 2007 | pmid = | pmc = | series = Lecture Notes in Computer Science | isbn = 978-3-540-72539-8 }}</ref> Ishai, Manoj Prabhakaran and Sahai[[Amit Sahai]]<ref>{{Cite book | last1 = Ishai | first1 = Y. | title = Advances in Cryptology – CRYPTO 2008 | last2 = Prabhakaran | first2 = M. | last3 = Sahai | first3 = A. | doi chapter=Founding 10.1007/978-3-540-85174-5_32Cryptography |on volumeOblivious =Transfer 5157– | pages = 572–591Efficiently | year = 2008 | pmid isbn= 978-3-540-85173-8 | pmc = | series = Lecture Notes in Computer Science |volume=5157 isbn |pages=572–591 |doi=10.1007/978-3-540-8517385174-8 5_32}}</ref> and Jesper Buus Nielsen and Claudio Orlandi.<ref>{{Cite book | last1 = Nielsen | first1 = J. B. | last2 = Orlandi | first2 = C. | doi = 10.1007/978-3-642-00457-5_22 | chapter = LEGO for Two-Party Secure Computation | title = Theory of Cryptography | series = Lecture Notes in Computer Science | volume = 5444 | pages = 368–386 | year = 2009 | isbn = 978-3-642-00456-8 | pmidciteseerx = 10.1.1.215.4422 }}</ref> Another solution for this problem, that explicitly works with committed input was proposed by Stanisław Jarecki and [[Vitaly Shmatikov]].<ref>{{Cite book | pmclast1 = Jarecki | citeseerxfirst1 = 10S.1 | title = Advances in Cryptology - EUROCRYPT 2007 | last2 = Shmatikov | first2 = V.1 | chapter = Efficient Two-Party Secure Computation on Committed Inputs | doi = 10.215.44221007/978-3-540-72540-4_6 | volume = 4515 | pages = 97–114 | year = 2007 | series = Lecture Notes in Computer Science | isbn = 978-3-540-72539-8 }}</ref>
Another solution for this problem, that explicitly works with committed input was proposed by Jarecki and Shmatikov.<ref>{{Cite book | last1 = Jarecki | first1 = S. | title = Advances in Cryptology - EUROCRYPT 2007 | last2 = Shmatikov | first2 = V. | doi = 10.1007/978-3-540-72540-4_6 | volume = 4515 | pages = 97–114 | year = 2007 | pmid = | pmc = | series = Lecture Notes in Computer Science | isbn = 978-3-540-72539-8 }}</ref>
== Secure multi-party computation ==
{{Main|Secure multi-party computation}}
==Security==
The security of a two-party computation protocol is usually defined through a comparison with an idealised scenario that is secure by definition.<ref>{{Cite journal |last1=Lindell |first1=Yehuda |last2=Pinkas |first2=Benny |title=An Efficient Protocol for Secure Two-Party Computation in the Presence of Malicious Adversaries |journal=Journal of Cryptology |year=2015 |language=en |volume=28 |issue=2 |pages=312–350 |doi=10.1007/s00145-014-9177-x |s2cid=253638839 |issn=0933-2790|doi-access=free }}</ref> The idealised scenario involves a [[Trusted third party|trusted party]] that collects the input of the two parties mostly client and server over [[secure channel]]s and returns the result if none of the parties chooses to abort.<ref>{{Citation |last1=Crépeau |first1=Claude |title=Statistical Security Conditions for Two-Party Secure Function Evaluation |date=2008 |url=http://link.springer.com/10.1007/978-3-540-85093-9_9 |work=Information Theoretic Security |volume=5155 |pages=86–99 |editor-last=Safavi-Naini |editor-first=Reihaneh |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-540-85093-9_9 |isbn=978-3-540-85092-2 |access-date=2022-10-19 |last2=Wullschleger |first2=Jürg|series=Lecture Notes in Computer Science |url-access=subscription }}</ref> The cryptographic two-party computation protocol is secure, if it behaves no worse than this ideal protocol, but without the additional [[trust (social sciences)|trust]] [[:wikt:assumption|assumptions]]. This is usually modeled using a simulator. The task of the simulator is to act as a wrapper around the idealised protocol to make it appear like the cryptographic protocol. The simulation succeeds with respect to an [[Information theory|information theoretic]], respectively [[computationally bounded adversary]] if the output of the simulator is [[statistically close]] to, respectively [[computationally indistinguishable]] from the output of the cryptographic protocol. A two-party computation protocol is secure, if for all adversaries there exists a successful simulator.
==See also==
* An important primitive in 2PC is [[oblivious transfer]].
* [[Universal composability]]
==References==
{{reflist}}
<references/>
[[Category:Cryptography]]
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