Secure multi-party computation (also known as secure computation or multi-party computation (MPC)) is a subfield of cryptography. The goal of this field is to create methods that enable parties to jointly compute a function over their inputs, while at the same time keeping these inputs private. For example, two millionaires can compute which one is richer, but without revealing their net worth. In fact, this very example was initially suggested by Andrew C. Yao in a 1982 paper., and was later named the millionaire problem. An excellent review of MPC is provided by Rashid Sheikh et al. in .
The concept is important in the field of cryptography and is closely related to the idea of zero knowledge. In general it refers to computational systems in which multiple parties wish to jointly compute some value based on individually held secret bits of information, but do not wish to reveal their secrets to one another in the process. For example, two individuals who each possess some secret information— and , respectively—may wish to jointly compute some function without revealing any information about and other than what can be reasonably deduced by knowing the actual value of , where "reasonably deduced" is often interpreted as equivalent to computation within polynomial time. The primary motivation for studying methods of secure computation is to design systems that allow for maximum utility of information without compromising user privacy.
The millionaire problem and its solution gave way to a generalization to multi-party protocols. In an MPC, a given number of participants p1, p2, ..., pN each have a private data, respectively d1, d2, ..., dN. The participants want to compute the value of a public function F on N variables at the point (d1, d2, ..., dN). An MPC protocol is secure if no participant can learn more from the description of the public function and the result of the global calculation than what he/she can learn from his/her own entry — under particular conditions depending on the model used.
It can be computational (i.e. based on some mathematical problem, like factoring) or unconditional (usually with some probability of error which can be made arbitrarily small).
The model in which the scheme is described might assume that participants use a synchronized network (a message sent at a "tick" always arrives at the next "tick"), that a secure and reliable broadcast channel exists, that a secure communication channel exists between every pair of participants (an adversary cannot read, modify or generate messages in the channel), etc.
The centrally controlled adversary considered can be passive (only allowed to read the data of a certain number of participants) or active (can corrupt the execution protocol or a certain number of participants).
An adversary can be static (chooses its victims before the start of the multi-party computation) or dynamic (can choose its victims during the course of execution of the multiparty computation). Attaining security against a dynamic adversary is often much harder than security against a static adversary.
An adversary can be defined as a threshold structure (meaning that it can corrupt or simply read the memory of a number of participants up to some threshold), or be defined as a more complex structure (it can affect certain predefined subsets of participants, modeling different possible collusions). These structures are commonly referred to as adversary structures. The opposite set consisting of the sets of honest parties that can still execute a computational task is related to the concept of access structures.
Unconditionally or information-theoretically secure MPC is closely related to the problem of secret sharing, and more specifically verifiable secret sharing (VSS); many secure MPC protocols that protect against active adversaries use VSS.
Performing a computation using MPC protocols is still orders of magnitude slower than performing the computation using a trusted third party. However, more and more efficient protocols for MPC have been proposed, and MPC can be now used as a practical solution to various real-life problems such as distributed voting, private bidding and auctions, sharing of signature or decryption functions, private information retrieval, etc. The first large-scale and practical application of multiparty computation took place in Denmark in January 2008, as described by Bogetoft et al.
The sub-problem of MPC that has received special attention by researchers because of its close relation to many cryptographic tasks is referred to as secure two-party computation (2PC) or just as Secure function evaluation (SFE). This area of research is concerned with the question: 'Can two party computation be achieved more efficiently and under weaker security assumptions than general MPC?'
Virtual Party Protocol
Virtual Party Protocol is an SMC protocol which uses virtual parties and complex mathematics to hide the identity of the parties.
Secure sum protocols
Secure sum protocols allow multiple cooperating parties to compute some function of their individual data without revealing the data to one another. Rashid Sheikh et al. proposed many secure sum protocols. 
^Pathak Rohit, Joshi Satyadhar, Advances in Information Security and Assurance, Springer Berlin / Heidelberg, ISSN 0302-9743 (Print) 1611-3349 (Online), ISBN 978-3-642-02616-4, DOI 10.1007/978-3-642-02617-1
^Rashid Sheikh, Brijesh Kumar and Durgesh Kumar Mishra, Privacy Preserving k-secure sum protocols, International Journal of Computer Science and Information Security, ISSN 1947-5500 (Online),Vol.6, No.2, Nov. 2009
Secure distributed CSP (DisCSP) solvers — a web-application with an applet-interpreter to design and run your own full-fledged secure multiparty computation (based on the SMC declarative language). Uses secure arithmetic circuit evaluation and mix-nets.
VMCrypt A Java library for scalable secure computation. By Lior Malka.
The Fairplay Project — Includes a software package for secure two-party computation, where the function is defined using a high-level function description language, and evaluated using Yao's protocol for secure evaluation of boolean circuits.
The SIMAP project; Secure Information Management and Processing (SIMAP) is a project sponsored by the Danish National Research Agency aimed implementing Secure Multiparty Computation.
MPCLib: Multi-Party Computation Library — A library written in C# and C++ that implements several building blocks required for implementing secure multi-party computation protocols. MPCLib has a discrete-event simulation engine that can be used for simulating MPC protocols in virtual networks.
MPC Java-based implementation A Java-based implementation of the MPC protocol based on Michael.B, Shafi.G and Avi.W's theorem ("Completeness theorems for non-cryptographic fault-tolerant distributed computation") with Welch-Berlekamp error correcting code algorithm to BCH codes. Supports multiple players and identification of "cheaters" with Byzantine protocol. By Erez Alon, Doron Friedland & Yael Smith.
SEPIA A java library for SMC using secret sharing. Basic operations are optimized for large numbers of parallel invocations (code available under the LGPL).