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.
We consider a minimal scenario for secure computation: Parties A and B have private inputs x and y and a shared random string r. A and B are each allowed to send a single message to a third party C, from which C is to learn the value of f(x, y) for some function f, but nothing else. We show that this model is surprisingly powerful: every function f can be securely computed in this fashion. If the messages are required to be of polynomial size, then we exhibit an efficient protocol for any function f computable in nondeterministic logspace. Using a computational notion of security, we exhibit efficient protocols for any polynomial-time computable function f, assuming the existence of one-way functions. The above results generalize to the case where there are more than two parties with private inputs. The minimalistic nature of our model makes it easy to transform positive results achieved in our model to other more general models of secure computation. It also gives hope for lowerbound proofs. We give an alternative characterization of our model in terms of graph embeddings, and use this to show that for most Boolean functions on {0, 1} n × {0, 1} n, the need to hide just one of the input bits from C requires a communication overhead of n bits. 1
Abstract We revisit the following open problem in information-theoretic cryptography: Does the communication complexity of unconditionally secure computation depend on the computational complexity of the function being computed? For instance, can computationally unbounded players compute an arbitrary function of their inputs with polynomial communication complexity and a linear threshold of unconditional privacy? Can this be done using a constant number of communication rounds? We provide an explanation for the difficulty of resolving these questions by showing that they are closely related to the problem of obtaining efficient protocols for (information-theoretic) private information retrieval and hence also to the problem of constructing short locally-decodable error-correcting codes. The latter is currently considered to be among the most intriguing open problems in complexity theory. Keywords. Information-theoretic cryptography, secure multiparty computation, private information retrieval, locally decodable codes. 1 Introduction In STOC 1990, Beaver, Micali, and Rogaway [5] posed the following question: Is there a constant-round protocol that allows k computationally unbounded players to defeat a computationally unbounded adversary, while using only a polynomial amount of communication in the total length of their inputs?
Abstract Card-based cryptography is a variant of multi-party computation using physical cards like playing cards. There are two models on card-based cryptography, called public and private models. The public model assumes that all operations are executed publicly, while the private model allows the players private operations called private permutations (PP, for short). Much of the existing card-based protocols were developed under the public model. Under the public model, 2ncards are necessary for every protocol withn-bit input since at least two cards are required to express a bit. In this paper, we proposen-bit input protocols with fewer than 2ncards by utilizing PP, which shows the power of PP. In particular, we show that a protocol for (n-bit input) threshold function can be realized with only
n+1 n+1 cards by reducing the threshold function to the majority voting. Toward this end, we first offer that two-bit input protocols for logic gates can be realized with fewer than four cards. Furthermore, we construct a new protocol for three-input majority voting with only four cards by observing the relationship between AND/OR operations. This protocol can be easily extended to more participants, and to the protocol for threshold functions.
Multiparty computation protocols (MPC) are said to be secure against covert adversaries if the honest parties are guaranteed to detect any misbehavior by the malicious parties with a constant probability. Protocols that, upon detecting a cheating attempt, additionally allow the honest parties to compute certificates, which enable third parties to be convinced of the malicious behavior of the accused parties, are called publicly verifiable. In this work, we make several contributions to the domain of MPC with security against covert adversaries. We identify a subtle flaw in a protocol of Goyal, Mohassel, and Smith (Eurocrypt 2008), meaning that their protocol does not allow to identify a cheating party, and show how to fix their original construction to obtain security against covert adversaries. We present generic compilers that transform arbitrary passively secure preprocessing protocols, i.e. protocols where the parties have no private inputs, into protocols that are secure against covert adversaries and publicly verifiable. Using our compiler, we construct the first efficient variants of the BMR and the SPDZ protocols that are secure and publicly verifiable against a covert adversary that corrupts all but one party, and also construct variants with covert security and identifiable abort. We observe that an existing impossibility result by Ishai, Ostrovsky, and Seyalioglu (TCC 2012) can be used to show that there exist certain functionalities that cannot be realized by parties, that have oracle-access to broadcast and arbitrary two-party functionalities, with information-theoretic security against a covert adversary.
The need for Cryptography arises out of the following fundamental question: can we perform useful computation while ensuring that an adversary does not learn anything about our private sensitive data? The notion of secure multiparty computation (MPC) \\cite{Yao82,GMW87} is a unifying framework for general secure protocols. MPC allows mutually distrusting parties to jointly evaluate any efficiently computable function on their private inputs in such a manner that each party does not learn anything beyond the output of the function. In this thesis, we study the question of building MPC protocols in various security models from standard cryptographic assumptions while minimizing the number of rounds of interaction amongst parties. In the first part of this thesis, (in a joint work with Vipul Goyal, Abhishek Jain, Yael Kalai, Dakshita Khurana and Amit Sahai, CRYPTO 2018) we construct the first round-optimal (i.e., four round) MPC protocol for general functions based on polynomially hard DDH (or QR or N
th-Residuosity) in the plain model where parties have access to no trusted setup. We further show how to overcome the four-round barrier for MPC by constructing a three-round protocol for ``list coin-tossing'' -- a slight relaxation of coin-tossing that suffices for most conceivable applications -- based on polynomially hard DDH (or QR or N
th-Residuosity). This result generalizes to randomized input-less functionalities. Previously, four round MPC protocols required sub-exponential-time hardness assumptions and no multi-party three-round protocols were known for any relaxed security notions with polynomial-time simulation against malicious adversaries. In order to build these protocols, we devise a new {\\em partitioned simulation} technique for MPC where the simulator uses different strategies for simulating the view of aborting adversaries and non-aborting adversaries. The protagonist of this technique is a new notion of {\\em promise zero knowledge} (ZK) where the ZK property only holds against non-aborting verifiers. We show how to realize promise ZK in three rounds in the simultaneous-message model assuming polynomially hard DDH (or QR or N
th-Residuosity). We also rely upon a new {\\em leveled rewinding security} technique that can be viewed as a polynomial-time alternative to leveled complexity leveraging for achieving ``non-malleability'' across different primitives. Then, we also we study the round complexity of concurrently secure multi-party computation (MPC) with super-polynomial simulation (SPS) in the plain model (in a joint work with Vipul Goyal, Abhishek Jain, Dakshita Khurana and Amit Sahai, TCC 2017). In the plain model, there are known explicit attacks that show that concurrently secure MPC with polynomial simulation is impossible to achieve; SPS security is the most widely studied model for concurrently secure MPC in the plain model. We construct a three-round concurrent MPC with SPS security against Byzantine adversaries, assuming sub-exponentially secure DDH and LWE. Prior to our work, the best known round complexity for SPS concurrent MPC was around twenty, although to the best of our knowledge, no previous work even gave an approximation of the constant round complexity that is sufficient for concurrent MPC.In the second part of the thesis, (in a joint work with Abhishek Jain, Rafail Ostrovsky and Ivan Visconti, ASIACRYPT 2018), we study the problem of non-interactive secure computation in the stateless hardware token model where parties have access to physical hardware as part of a trusted setup phase. The notion of non-interactive secure computation (NISC) first introduced in the work of Ishai et al. [EUROCRYPT 2011] studies the following problem: Suppose a receiver
R wishes to publish an encryption of her secret input
y so that any sender
S with input
x can then send a message
m that reveals
f(x,y) to
R (for some function
f). Here,
m can be viewed as an encryption of
f(x,y) that can be decrypted by
R. NISC requires security against both malicious senders and receivers, and also requires the receiver's message to be reusable across multiple computations (w.r.t. a fixed input of the receiver). All previous solutions to this problem necessarily rely upon OT (or specific number-theoretic assumptions) even in the common reference string model or the random oracle model or to achieve weaker notions of security such as super-polynomial-time simulation. In this work, we construct a NISC protocol based on the minimal assumption of one way functions, in the stateless hardware token model. Our construction achieves UC security and requires a single token sent by the receiver to the sender.