Secure multi-party computation (also known as secure computation, multi-party computation (MPC) or privacy-preserving computation) is a subfield of cryptography with the goal of creating methods for parties to jointly compute a function over their inputs while keeping those inputs private. Unlike traditional cryptographic tasks, where cryptography assures security and integrity of communication or storage and the adversary is outside the system of participants (an eavesdropper on the sender and receiver), the cryptography in this model protects participants' privacy from each other.
The foundation for secure multi-party computation started in the late 1970s with the work on mental poker, cryptographic work that simulates game playing/computational tasks over distances without requiring a trusted third party. Traditionally, cryptography was about concealing content, while this new type of computation and protocol is about concealing partial information about data while computing with the data from many sources, and correctly producing outputs. By the late 1980s, Michael Ben-Or, Shafi Goldwasser and Avi Wigderson, and independently David Chaum, Claude Crépeau, and Ivan Damgård, had published papers showing "how to securely compute any function in the secure channels setting".
History Special purpose protocols for specific tasks started in the late 1970s.A. Shamir, R. Rivest, and L. Adleman, "Mental Poker", Technical Report LCS/TR-125, Massachusetts Institute of Technology, April 1979. Later, secure computation was formally introduced as secure two-party computation (2PC) in 1982 (for the so-called Millionaires' Problem, a specific problem which is a Boolean predicate), and in generality (for any feasible computation) in 1986 by Andrew Yao.Andrew C. Yao, Protocols for secure computations (extended abstract)Andrew Chi-Chih Yao:How to Generate and Exchange Secrets (Extended Abstract). FOCS 1986: 162-167 The area is also referred to as Secure Function Evaluation (SFE). The two party case was followed by a generalization to the multi-party by Oded Goldreich, Silvio Micali, and Avi Wigderson. The computation is based on secret sharing of all the inputs and zero-knowledge proofs for a potentially malicious case, where the majority of honest players in the malicious adversary case assure that bad behavior is detected and the computation continues with the dishonest person eliminated or his input revealed. This work suggested the very basic general scheme to be followed by essentially all future multi-party protocols for secure computing.Oded Goldreich, Silvio Micali, Avi Wigderson:How to Play any Mental Game or A Completeness Theorem for Protocols with Honest Majority. STOC 1987: 218-229 This work introduced an approach, known as GMW paradigm, for compiling a multi-party computation protocol which is secure against semi-honest adversaries to a protocol that is secure against malicious adversaries. This work was followed by the first robust secure protocol which tolerates faulty behavior graciously without revealing anyone's output via a work which invented for this purpose the often used `share of shares idea'Zvi Galil, Stuart Haber, Moti Yung: Cryptographic Computation: Secure Fault-Tolerant Protocols and the Public-Key Model. CRYPTO 1987: 135-155 and a protocol that allows one of the parties to hide its input unconditionally.David Chaum, Ivan Damgård, Jeroen van de Graaf: Multiparty Computations Ensuring Privacy of Each Party's Input and Correctness of the Result. 87-119 The GMW paradigm was considered to be inefficient for years because of huge overheads that it brings to the base protocol. However, it is shown that it is possible to achieve efficient protocols, and it makes this line of research even more interesting from a practical perspective. The above results are in a model where the adversary is limited to polynomial time computations, and it observes all communications, and therefore the model is called the `computational model'. Further, the protocol of oblivious transfer was shown to be complete for these tasks.Joe Kilian: Founding Cryptography on Oblivious Transfer. STOC 1988: 20-31 The above results established that it is possible under the above variations to achieve secure computation when the majority of users are honest.
The next question to solve was the case of secure communication channels where the point-to-point communication is not available to the adversary; in this case it was shown that solutions can be achieved with up to 1/3 of the parties being misbehaving and malicious, and the solutions apply no cryptographic tools (since secure communication is available).Michael Ben-Or, Shafi Goldwasser, Avi Wigderson: Completeness Theorems for Non-Cryptographic Fault-Tolerant Distributed Computation (Extended Abstract). STOC 1988: 1-10 Adding a broadcast channel allows the system to tolerate up to 1/2 misbehaving minority,Tal Rabin, Michael Ben-Or: Verifiable Secret Sharing and Multiparty Protocols with Honest Majority (Extended Abstract). STOC 1989: 73-85 whereas connectivity constraints on the communication graph were investigated in the book Perfectly Secure Message Transmission.Danny Dolev, Cynthia Dwork, Orli Waarts, Moti Yung: Perfectly Secure Message Transmission. J. ACM 40(1): 17-47 (1993)
Over the years, the notion of general purpose multi-party protocols became a fertile area to investigate basic and general protocol issues properties on, such as universal composability or mobile adversary as in proactive secret sharing.Rafail Ostrovsky, Moti Yung: How to Withstand Mobile Virus Attacks. PODC 1991. pp. 51-59
Since the late 2000s, and certainly since 2010 and on, the domain of general purpose protocols has moved to deal with efficiency improvements of the protocols with practical applications in mind. Increasingly efficient protocols for MPC have been proposed, and MPC can be now considered as a practical solution to various real-life problems (especially ones that only require linear sharing of the secrets and mainly local operations on the shares with not much interactions among the parties), such as distributed voting, private bidding and auctions, sharing of signature or decryption functions and private information retrieval.Claudio Orlandi: Is multiparty computation any good in practice?, ICASSP 2011 The first large-scale and practical application of multi-party computation was the execution of an electronic double auction in the Danish Sugar Beet Auction, which took place in January 2008. Obviously, both theoretical notions and investigations, and applied constructions are needed (e.g., conditions for moving MPC into part of day by day business was advocated and presented inMoti Yung: From Mental Poker to Core Business: Why and How to Deploy Secure Computation Protocols? ACM Conference on Computer and Communications Security 2015: 1-2 https://dl.acm.org/citation.cfm?doid=2810103.2812701).
In 2020, a number of companies working with secure-multiparty computation founded the MPC alliance with the goal of "accelerate awareness, acceptance, and adoption of MPC technology."
Definition and overview
In an MPC, a given number of participants, p1, p2, ..., pN, each have private data, respectively d1, d2, ..., dN. Participants want to compute the value of a public function on that private data: F(d1, d2, ..., dN) while keeping their own inputs secret.
For example, suppose we have three parties Alice, Bob and Charlie, with respective inputs x, y and z denoting their salaries. They want to find out the highest of the three salaries, without revealing to each other how much each of them makes. Mathematically, this translates to them computing:
If there were some trusted outside party (say, they had a mutual friend Tony who they knew could keep a secret), they could each tell their salary to Tony, he could compute the maximum, and tell that number to all of them. The goal of MPC is to design a protocol, where, by exchanging messages only with each other, Alice, Bob, and Charlie can still learn without revealing who makes what and without having to rely on Tony. They should learn no more by engaging in their protocol than they would learn by interacting with an incorruptible, perfectly trustworthy Tony.
In particular, all that the parties can learn is what they can learn from the output and their own input. So in the above example, if the output is , then Charlie learns that his is the maximum value, whereas Alice and Bob learn (if , and are distinct), that their input is not equal to the maximum, and that the maximum held is equal to . The basic scenario can be easily generalised to where the parties have several inputs and outputs, and the function outputs different values to different parties.
Informally speaking, the most basic properties that a multi-party computation protocol aims to ensure are: Input privacy: No information about the private data held by the parties can be inferred from the messages sent during the execution of the protocol. The only information that can be inferred about the private data is whatever could be inferred from seeing the output of the function alone. Correctness: Any proper subset of adversarial colluding parties willing to share information or deviate from the instructions during the protocol execution should not be able to force honest parties to output an incorrect result. This correctness goal comes in two flavours: either the honest parties are guaranteed to compute the correct output (a "robust" protocol), or they abort if they find an error (an MPC protocol "with abort").
There are a wide range of practical applications, varying from simple tasks such as coin tossing to more complex ones like electronic auctions (e.g. compute the market clearing price), electronic voting, or privacy-preserving data mining. A classical example is the Millionaires' Problem: two millionaires want to know who is richer, in such a way that neither of them learns the net worth of the other. A solution to this situation is essentially to securely evaluate the comparison function.
Security definitions
A multi-party computation protocol must be secure to be effective. In modern cryptography, the security of a protocol is related to a security proof. The security proof is a mathematical proof where the security of a protocol is reduced to that of the security of its underlying primitives. Nevertheless, it is not always possible to formalize the cryptographic protocol security verification based on the party knowledge and the protocol correctness. For MPC protocols, the environment in which the protocol operates is associated with the Real World/Ideal World Paradigm.Michael Backes, Birgit Pfitzmann, and Michael Waidner. "A general composition theorem for secure reactive systems." In Theory of Cryptography Conference, pp. 336-354. Springer, Berlin, Heidelberg, 2004. The parties can't be said to learn nothing, since they need to learn the output of the operation, and the output depends on the inputs. In addition, the output correctness is not guaranteed, since the correctness of the output depends on the parties’ inputs, and the inputs have to be assumed to be correct.
The Real World/Ideal World Paradigm states two worlds: (i) In the ideal-world model, there exists an incorruptible trusted party to whom each protocol participant sends its input. This trusted party computes the function on its own and sends back the appropriate output to each party. (ii) In contrast, in the real-world model, there is no trusted party and all the parties can do is to exchange messages with each other. A protocol is said to be secure if one can learn no more about each party's private inputs in the real world than one could learn in the ideal world. In the ideal world, no messages are exchanged between parties, so real-world exchanged messages cannot reveal any secret information.