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.
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.
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.
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) considers the problem where a set of mutually distrusting parties want to jointly compute a function over their private inputs, without revealing any extra information about these inputs other than what it can be inferred from the output of the function. This setting is well-motivated, and it has many real-world applications such as auction, voting, etc. MPC can be also seen as a generalization of many natural cryptographic primitives. For example, zero-knowledge (ZK) can be viewed as a special case of two-party secure computation. In ZK, a party, called prover aims to convince a second party, called verifier, that the prover’s private input witness w and a public input statement x belong to a relation R. An important research direction in secure computation is to find the trade-off between the required setup (e.g., the use of the broadcast channel, the use of common reference string (CRS) / public key infrastructure (PKI), the upper bound of the parties that can be corrupted, etc.), and the security guarantees that can be achieved. The setups can be viewed as some general assumptions that the protocol needs to satisfy, and they influence the usability of the protocol in real-world scenarios. In principle, having simpler (or no) setups mean that the protocol is more general and can be more useful in real-world scenarios. At the same time, having simpler setups may lead to weaker security guarantees. Therefore, finding the trade-off between setup and security guarantees is important and meaningful. In this thesis, we target MPC and ZK, and we focus on how to minimize the setup for MPC and ZK while still providing meaningful levels of security. More specifically: Regarding MPC, we focus on the dishonest majority (i.e., the adversary can corrupt all but one party), and we aim at 1) minimizing the use of broadcast channels. 2) studying the MPC with pre-processing when no setup is available. • Informally, a broadcast channel guarantees that when a message is sent, this reaches all the parties, without ambiguity. It also guarantees that if an honest party receives a message from a corrupted party, then it is guaranteed that all the honest parties have received the message. To realize broadcast, parties in the protocol could run the broadcast protocol, which may require many rounds of peer-to-peer communications. An alternative way is to rely on physical or external infrastructure such as blockchain. In both cases, broadcast is expensive, as such, we want to minimize its use. In particular, this thesis presents the following results: – When assuming no setup, we give a complete characterization with respect to the use of broadcast channels, and we obtain the optimal results. – We consider the same problem for the case that we only want to allow the black-box use (i.e., do not have access to the code of the algorithm) of the oblivious transfer protocol. We also give a characterization. • In the standard definition of MPC, the parties’ private inputs are fixed before the start of the protocol. However, there is another type of MPC named MPC with pre-processing, where the protocol can pre-compute some messages without using parties’ inputs, and these messages can accelerate computations in the online phase (i.e., other computations that require parties’ inputs). Since some expensive computations can be pre-computed, the online phase could be more lightweight. Therefore, we want to remove the dependency of the input from as many rounds as possible, so that we can do some pre-processing. In this direction, we explore the protocol with no setup. We provide a compiler that can turn a big class of MPC protocol that may require the inputs already to compute the first round, into a new protocol that needs the inputs only in the last two rounds. We also propose new MPC definitions that capture this delayed-input features. Regarding ZK, we do the following: • In standard single-theorem ZK definition, the security of the ZK protocol is guaranteed to hold only when one proof is issued. In the case where multiple zero-knowledge proofs need to be issued (i.e., to prove multiple NP statements), each new zero-knowledge proof requires a freshly generated setup. In the multi-theorem ZK definition, instead, one setup is sufficient for generating multiple zero-knowledge proofs for multiple instances. We propose a multi-theorem protocol (in the format of a compiler) that follows the Fiat-Shamir paradigm and relies on correlation intractable hash functions. Moreover, our protocol remains zero-knowledge and sound even against adversaries that choose the statement to be proven (and the witness for the case of zero-knowledge) adaptively on the key of the hash function. Prior works could achieve this adaptive security only inefficiently via NP reductions. • ZK protocols are secure only when all setups are correctly generated, but in real-world scenarios, some of the setups may not be correctly generated. For instance, to run a non-interactive zero-knowledge (NIZK) protocol, the setup CRS could be chosen with bias. In this case, the security of the NIZK protocol does not hold anymore. Instead of finding a secure ZK candidate, one alternative solution is to have multiple instantiations of ZK candidates and assume that only for a subset of them the setup is generated correctly. More formally, we consider the case where only a subset of the instances are secure. In more detail, given access to n candidate instantiations of a NIZK for some language, we want to have a construction that itself implements a NIZK for the same language without relying on any additional computational assumptions. We refer to this type of construction as combiner, and the combiner is secure assuming at least t of the given candidates are secure. In this work, we provide three different constructions of robust NIZK combiners and show that combiners are impossible to realize unless the majority of the input candidates are secure.
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 Nth-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 Nth-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 Nth-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.
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.
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.