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Non-Interactive Secure Multiparty Computation∗

Amos Beimel, Ariel Gabizon, Yuval Ishai, Eyal Kushilevitz Z, Sigurd Meldgaard, Anat Paskin-Cherniavsky

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

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-

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•Jan 1, 2012•2 citations
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) enables multiple players to cooperatively evaluate various functions in the presence of adversaries. In this paper, we consider non-interactive MPC (NIMPC) against honest-but-curious adversaries in the information-theoretic setting, which was introduced by Beimel et al. at CRYPTO 2014. Their main focus is to realize stronger security while completely avoiding interaction, and succeeded to show that every function admits a fully robust NIMPC protocol. In this paper, we further develop the study of NIMPC. We first present a simple lower bound on the communication complexity derived from the correctness requirement of NIMPC. Secondly, we present an efficient NIMPC protocol for indicator functions, which is an important building block of NIMPC protocols. An NIMPC protocol for arbitrary functions is also constructed from the proposed NIMPC for indicator functions by using the generic compiler introduced by Beimel et al. in CRYPTO 2014. The communication complexities of NIMPC protocols presented in this paper are much more efficient than the previous ones. In fact, the gap between the lower and upper bounds of the communication complexity is reduced from exponential in the input length to quadratic . Finally, we show some improvements on the efficiency in the so-called offline-online model. Specifically, for some sets of functions, the exponential amount of offline communication reduces the online communication to almost optimum amount in the standard model.
Non-interactive secure multiparty computation is a method to share and compute a private functionfffamongnnnparties holding private inputs xi with 1nnnin the information theoretical setting. Each party locally computes a message from a share offffand send it to a referee in parallel. The referee obtains the output f (xi,… , xn) while keepingfffand xi private other than what can be inferred from the output. The communication complexity is evaluated by the maximum length of shares and messages. In the previous work, several kinds of function families, upper and lower bounds have been improved and the only gaps remaining are for the group products and the Abelian programs (including symmetric functions). This paper improves the communication complexity of both function families towards the optimum by deriving a new lower bound for the group products that matches the previous upper bound (up to a constant factor) and the first asymptotically optimal upper bound for the Abelian programs.
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 notion of secure computation is central to cryptography. Introduced in the seminal works of Yao [FOCS'82, FOCS'86] and Goldreich, Micali and Wigderson [STOC'87], secure multiparty computation allows a group of (mutually) distrustful parties to jointly compute any functionality over their individual private inputs in such a manner that the honest parties obtain the correct outputs and no group of malicious parties learn anything beyond their inputs and the prescribed outputs. General feasibility results for secure computation were given by Yao [FOCS'86] and Goldreich et al. [STOC'87] more than two decades ago. Subsequent to these works, designing secure computation protocols that can tolerate more powerful adversaries and satisfy stronger notions of security has been a very active area of research. In this dissertation, we study two such new frontiers in the area of secure computation. In the first part of this dissertation, we initiate a study of designing leakage-resilient interactive protocols. Specifically, we consider the scenario where an adversary, in addition to corrupting a subset of parties, can leak (potentially via physical attacks) arbitrary information from the secret state of any honest party. This is in contrast to the standard notion of secure computation where it is assumed that the adversary only has black-box'' access to the honest parties. In particular, we formalize a meaningful definition of leakage-resilient zero knowledge proof systems and provide constructions that satisfy our notion. We also discuss various applications of our results. The second part of this dissertation concerns with the general question of whether it is possible to securely run cryptographic protocols over an insecure network environment such as the Internet. It is well-known that the standard notion of secure computation is only relevant to the stand-alone setting where a single protocol is being executed in isolation; as such it does not guarantee security when multiple protocol sessions may be executed concurrently under the control of an adversary who is present across all sessions. We consider the open problem of constructing secure password-based authenticated key exchange protocols in such a setting in the plain model (i.e., without assuming any trusted infrastructure or random oracles). We give the first construction of such a protocol based on standard cryptographic assumptions. Our results are in fact much more general, and extend to other functionalities w.r.t. a (necessarily) weakened notion of concurrently secure computation. The results presented in this dissertation stem from two papers which are joint works with Sanjam Garg and Amit Sahai, and with Vipul Goyal and Rafail Ostrovsky, respectively.
A fundamental problem in the theory of secure multi-party computation (MPC) is to characterize functions with more than 2 parties which admit MPC protocols with information-theoretic security against passive corruption. This question has seen little progress since the work of Chor and Ishai (2001), which demonstrated difficulties in resolving it. In this work, we make significant progress towards resolving this question in the important case of aggregating functionalities, in which m parties P1,...,Pm hold inputs x1,...,xm and an aggregating party P0 must learn f(x1,...,xm). We give a necessary condition and a slightly stronger sufficient condition for f to admit a secure protocol. Both the conditions are stated in terms of an algebraic structure we introduce called Commuting Permutations Systems (CPS), which may be of independent combinatorial interest. When our sufficiency condition is met, we obtain a perfectly secure protocol with minimal interaction, that fits the model of Non-Interactive MPC or NIMPC (Beimel et al., 2014), but without the need for a trusted party to generate correlated randomness. We define Unassisted Non-Interactive MPC (UNIMPC) to capture this variant. We also present an NIMPC protocol for all functionalities, which is simpler and more efficient than the one given in the prior work.
In the setting of secure multiparty computation, a set of parties wish to compute a joint function of their inputs, while preserving properties like privacy, correctness, and independence of inputs. One security property that has typically not been considered in the past relates to the length or size of the parties inputs. This is despite the fact that in many cases the size of a party’s input can be confidential. The reason for this omission seems to have been the folklore belief that, as with encryption, it is impossible to carry out non-trivial secure computation while hiding the size of parties ’ inputs. However some recent results (e.g., Ishai and Paskin at TCC 2007, Ateniese, De Cristofaro and Tsudik at PKC 2011) showed that it is possible to hide the input size of one of the parties for some limited class of functions, including secure two-party set intersection. This suggests that the folklore belief may not be fully accurate. In this work, we initiate a theoretical study of input-size hiding secure computation, and focus on the two-party case. We present definitions for this task, and deal with the subtleties that arise in the setting where there is no a priori polynomial bound on the parties ’ input sizes. Our definitional study yields a multitude of classes of input-size hiding computation, depending