Background: secure multiparty computation

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

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). Adding a broadcast channel allows the system to tolerate up to 1/2 misbehaving minority, whereas connectivity constraints on the communication graph were investigated in the book Perfectly Secure Message Transmission.

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.

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.

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 in).

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.

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.

The Real World/Ideal World Paradigm provides a simple abstraction of the complexities of MPC to allow the construction of an application under the pretense that the MPC protocol at its core is actually an ideal execution. If the application is secure in the ideal case, then it is also secure when a real protocol is run instead.

Protocols

There are major differences between the protocols proposed for two party computation (2PC) and multi-party computation (MPC). Also, often for special purpose protocols of importance a specialized protocol that deviates from the generic ones has to be designed (voting, auctions, payments, etc.)