Peter Bogetoft, Dan Lund Christensen, Ivan Damgård, Martin Geisler, Thomas Pelle Jakobsen, Mikkel Krøigaard, Janus Dam Nielsen, Jesper Buus Nielsen, Kurt Nielsen, Jakob Illeborg Pagter, Michael I. Schwartzbach, Tomas Toft
Abstract. In this note, we report on the first large-scale and practical application of multiparty computation, which took place in January 2008. We also report on the novel cryptographic protocols that were used. 1 Introduction and History In multiparty computation (MPC), we consider a number of players P1,..., Pn, who initially each hold inputs x1,..., xn, and we then want to securely compute some function f on these inputs, where f(x1,..., xn) = (y1,..., yn), such that Pi learns yi but no other information. This should hold, even if players exhibit some amount of adversarial behavior. The goal can be accomplished by an interactive
Abstract. In this note, we report on the first large-scale and practical application of multiparty computation, which took place in January 2008. We also report on the novel cryptographic protocols that were used. 1 Introduction and History In multiparty computation (MPC), we consider a number of players P1,..., Pn, who initially each hold inputs x1,..., xn, and we then want to securely compute some function f on these inputs, where f(x1,..., xn) = (y1,..., yn), such that Pi learns yi but no other information. This should hold, even if players exhibit some amount of adversarial behavior. The goal can be accomplished by an interactive
Secure Multiparty Computation (abrv. MPC) is a group of techniques that enable multiple entities to compute some function on their private inputs. Moreformally,itenablesasetofplayers{P1,...,Pn},eachofthemholding private inputs xi, where i is the index of the player, to evaluate a function f(x1,...,xn) = (y1,...,yn) so that every player Pi learns yi. Multiparty Computation has many application scenarios such as private auctioning, computation outsourcing, or private information retrieval. These and other applications make MPC a very appealing object of study, since it may be a solution to problems in different fields. The objective of this thesis is twofold. First, we aim to provide a comprehensive guide to the current state of multiparty computation protocols. Wedosobyinspectingthetwocategorieswheremostgeneralfunctionality secure function evaluation protocols are inserted - boolean or arithmetic circuit evaluation. Moreover, we show how to implement MPC protocols with current constructions, and what problems are inherently connected to the chosen representations. Second, we document some of the design choices behind a partial implementation of a concrete secure arithmetic circuit evaluation protocol - the SPDZ protocol by Damgård et al. [10]. We explain the protocol in general terms, and then go into the details of some subprotocols, namely those that were implemented.
The aim of this paper is to present some of the recent progress in efficient secure multiparty computation (MPC). In MPC we have a set of parties owning a set of private inputs. The par ties want to compute a function of their inputs, but they do not trust each other, therefore they need a cryptographic protocol to perform the computation in a way that 1) the output is correct and 2) cheating parties will not be able to learn any information about the honest parties inputs. Even though this problem has been formulated and essentially solved almost 30 years ago, practical solutions that can be relevant for real-world applications have been discovered only in the last few years. We will present some of these advances, trying to explain to a non-specialized audience the significance of the several existing security notions.
We introduce and study the notion of non-interactive secure multiparty computation (NIMPC). An NIMPC protocol for a function f(x1,..., xn) is specified by a joint probability distribution R = (R1,..., Rn) and local encoding functions Enci(xi, Ri), 1 ≤ i ≤ n. Given correlated randomness (R1,..., Rn) ∈R R, each party Pi, using its input xi and its randomness Ri, computes the message mi = Enci(xi, Ri). The messagesm1,...,mn can be used to decode f(x1,..., xn). For a set T ⊆ [n], the protocol is said to be T-robust if revealing the messages (Enci(xi, Ri))i 6∈T together with the random-ness (Ri)i∈T gives the same information about (xi)i 6∈T as an oracle access to the function f restricted to these input values. Namely, a coalition T can learn no more than the restriction of f fixing the inputs of uncorrupted parties, which, in this non-interactive setting, one cannot hope to hide. For 0 ≤ t ≤ n, the protocol is t-robust if it is T-robust for every T of size at most t and it is fully robust if it is n-robust. A 0-robust NIMPC protocol for f coincides with a protocol in the private simultaneous messages model of Feige et al. (STOC 1994). In the setting of computational (indistinguishability-based) security, fully robust NIMPC is implied by multi-input functional encryption, a notion that was recently introduced by Goldwasser et al. (Euro-
Secure multi-party computation (MPC) is a cryptographic primitive that lets a num- ber N of mutually distrustful parties compute a function f (x1, . . . , xN ) on their private inputs xi such that at the end of the MPC protocol, each honest party obtains the func- tion’s correct output and no adversary controlling a certain subset of parties learns any- thing about the honest parties’ inputs beyond what can be inferred from the function’s output value. In 2002, [Smi01] and [CGS02] introduced the notion of multi-party quan- tum computation (MPQC) in which arbitrary quantum circuits can be evaluated in a distributed manner, secure against a quantum adversary. Few protocols for MPQC are known. We present a fundamental study of the building blocks of MPC and MPQC in the information-theoretic setting with a focus on protocols built on top of (quantum) se- cret sharing schemes. In particular, we point out structural similarities and differences and compile an extensive list of theoretical feasibility results for the maximum number of corrupted parties within classical and quantum secret sharing and MPC protocols.
Introduction In Chapter 1 we gave an introduction to the multiparty computation problem and explained what it intuitively means for a protocol to compute a given function securely, namely, we want a protocol that is correct and private. However, we only considered three players, we only argued that a single player does not learn more than he or she is supposed to, and finally, we assumed that all players would follow the protocol. In this chapter we will consider a more general solution to multiparty computation, where we remove the first two restrictions: we will consider any number n ≥ 3 of players, and we will be able to show that as long as at most t < n /2 of the players go together after the protocol is executed and pool all their information, they will learn nothing more than their own inputs and the outputs they were supposed to receive, even if their computing power is unbounded. We will still assume, however, that all players follow the protocol. This is known as semihonest or passive security . To argue security in this chapter, we will use a somewhat weak but very simple definition that only makes sense for semihonest security. We then extend this in Chapter 4 to a fully general model of what protocols are and what security means. Throughout this chapter we will assume that each pair of players can communicate using a perfectly secure channel, so if two players exchange data, a third player has no information at all about what is sent. Such channels might be available because of physical circumstances, or we can implement them relative to a computational assumption using cryptography. For more details on this, see Chapter 7. We end the chapter by showing that the bound t < n /2 is optimal: if t ≥ n /2, then some functions cannot be computed securely in the model mentioned, that is, obtaining perfect security by using only secure point-to-point communication as the communication resource. Secret Sharing Our main tool to build the protocol will be secret-sharing schemes. The theory of secret-sharing schemes is a large and interesting field in its own right with many applications to multiparty computation (MPC), and we look at this in more detail in Chapter 11, where a formal definition of the notion as well as a general treatment of the theory of secret sharing can be found.
Secure Multiparty Computation (MPC) is an expression encompassing cryptographic techniques that allow mutually distrustful parties to jointly perform computations over private data, ensuring correctness and preserving privacy despite malicious behavior. Guaranteed output is the strongest security goal for MPC: protocols with this property always deliver results. This is in contrast with weaker notions that allow malicious aborts. The seminal works of Ben-Or, Goldwasser, and Wigderson (STOC’88) and Rabin and Ben-Or (STOC’89) showed interactive protocols to compute any function with guaranteed output and perfect security if fewer than one third of parties are corrupted, or with statistical security (with a small error probability) if fewer than half are corrupted. These results hold when the set of participants is fixed throughout the execution. Ostrovsky and Yung’s proactive model (PODC’91) relaxes this assumption by allowing the set of corrupted parties to evolve over time, better capturing the dynamics of long-lived computations, where machines may be periodically compromised and recovered. The recent YOSO (You Only Speak Once) and fluid MPC models (CRYPTO’21) push this ambitious modelling even further by allowing for fully dynamic participation, with parties joining and leaving the computation at will. In this setting, known protocols with guaranteed output fall short of optimal resilience or rely on non-standard communication and adversarial assumptions. Matching the classical BGW and Rabin–Ben-Or bounds in these settings remained an open problem. We resolve these questions within the Layered MPC abstraction of David et al. (CRYPTO’23), in which the interaction pattern of a protocol is restricted to be a layered graph, meaning parties are partitioned into a sequence of disjoint sets (layers) and can send private and broadcast messages to parties in immediately subsequent layers. We match the BGW and Rabin and Ben-Or bounds in this model, fully characterizing the feasibility landscape of Layered MPC with guaranteed output. As a stepping stone towards our statistically secure protocol, we construct a more efficient and conceptually simpler computationally secure protocol relying on linear non-interactive commitments. Our results yield optimally-resilient and maximally-proactive MPC protocols with guaranteed output for both perfect and statistical security.