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International Conference on Distributed Computing and Networking (ICDCN)

Optimally Resilient Asynchronous MPC with Linear Communication Complexity

Ashish Choudhury, Arpita Patra

Published Jan 4, 2015• 30 citations
Publisher

Abstract

We present a secure asynchronous multiparty computation (AMPC) protocol with optimal resilience, involving n = 3t + 1 parties and tolerating a computationally bounded static adversary, capable of corrupting upto t parties. For a security parameter k and for circuits of sufficiently large size, our protocol has an amortized communication complexity of O(cMnk) bits, where cM denotes the number of multiplication gates in the arithmetic circuit, representing the function to be computed. Prior to our work, the most efficient optimally resilient, computationally secure AMPC protocol was due to Hirt et al. (ICALP 2008). The protocol offers an amortized communication complexity of O(cMn2k) bits.

DOI
10.1145/2684464.2684470

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We present a secure asynchronous multiparty computation (AMPC) protocol with optimal resilience, involving n = 3t + 1 parties and tolerating a computationally bounded static adversary, capable of corrupting upto t parties. For a security parameter k and for circuits of sufficiently large size, our protocol has an amortized communication complexity of O(cMnk) bits, where cM denotes the number of multiplication gates in the arithmetic circuit, representing the function to be computed. Prior to our work, the most efficient optimally resilient, computationally secure AMPC protocol was due to Hirt et al. (ICALP 2008). The protocol offers an amortized communication complexity of O(cMn2k) bits.
In this paper, we present an optimally-resilient, unconditionally-secure asynchronous multi-party computation (AMPC) protocol for n parties, tolerating a computationally unbounded adversary, capable of corrupting up to t < n/3 parties. Our protocol needs a communication of 𝒪(n⁴) field elements per multiplication gate. This is to be compared with previous best AMPC protocol (Patra et al, ICITS 2009) in the same setting, which needs a communication of 𝒪(n⁵) field elements per multiplication gate. To design our protocol, we present a simple and highly efficient asynchronous verifiable secret-sharing (AVSS) protocol, which is of independent interest.
Unconditionally secure multiparty computation (MPC) allows a set of n mutually distrusting parties to securely compute an agreed function f over some finite field in the presence of a computationally unbounded adversary, who can actively corrupt any t out of the n parties. Designing an asynchronous MPC (AMPC) protocol with a communication complexity of O(n) field elements per multiplication gate is a long standing open problem. We solve the open problem by presenting two AMPC protocols with the corruption threshold t < n/4. Our first protocol is statistically secure (i.e. involves a negligible error in the protocol outcome) in a completely asynchronous setting and improves the communication complexity of the previous best AMPC protocol in the same setting by a factor of Θ(n). Our second protocol is perfectly secure (i.e. error free) in a hybrid setting, where one round of communication is assumed to be synchronous and improves the communication complexity of the previous best AMPC protocol in the hybrid setting by a factor of Θ(n 2). Starting with the seminal work of Beaver (CRYPTO 1991), it is by now a well-known technique to evaluate multiplication gates in an MPC protocol using shared random multiplication triples. The central contribution common to both the presented protocols is a new and simple framework for generating shared random multiplication triples. All the existing protocols approach the problem by first producing shared pairs of random values, followed by computing
In this paper, we design secure multi-party computation (MPC) protocols in the asynchronous communication setting with optimal resilience. Our protocols are secure against a computationally-unbounded malicious adversary, characterized by an adversary structure Z\mathcal{Z}Z, which enumerates all possible subsets of potentially corrupt parties. Our protocols incur a communication of O(∣Z∣2)\mathcal{O}(|\mathcal{Z}|^2)O(∣Z∣2) and O(∣Z∣)\mathcal{O}(|\mathcal{Z}|)O(∣Z∣) bits per multiplication for perfect and statistical security respectively. These are the first protocols with this communication complexity, as such protocols were known only in the synchronous communication setting (Hirt and Tschudi, ASIACRYPT 2013).
Multiparty computation (MPC) among n parties can tolerate up to t < n/2 active corruptions in a synchronous communication setting; however, in an asynchronous communication setting, the resiliency bound decreases to only t < n/3 active corruptions. We improve the resiliency bound for asynchronous MPC (AMPC) to match synchronous MPC using non-equivocation. Non-equivocation is a message authentication mechanism to restrict a corrupted sender from mak-ing conflicting statements to different (honest) parties. It can be implemented using an increment-only counter and a digital signature oracle, realizable with trusted hardware modules readily available in com-modity computers and smartphone devices. A non-equivocation mechanism can also be transferable and allow a receiver to verifiably transfer the authenticated statement to other parties. In this work, using transferable non-equivocation, we present an AMPC protocol tolerating t < n/2 faults. From a practical point of view, our AMPC protocol requires fewer setup assumptions than the previous AMPC protocol with t < n/2 by Beerliová-Trubı́niová, Hirt and Nielsen [PODC 2010]: unlike their AMPC protocol, it does not require any synchronous broadcast round at the beginning of the protocol and avoids the threshold homomorphic encryption setup assumption. Moreover, our AMPC protocol is also efficient and provides a gain of Θ(n) in the communication complexity per multiplication gate, over the AMPC protocol of Beerliová-Trubı́niova ́ et al. In the process, using non-equivocation, we also define the first asynchronous verifiable secret sharing (AVSS) scheme with t < n/2, which is of independent interest to threshold cryptography. 1
We describe scalable protocols for solving the secure multi-party computation (MPC) problem among a large number of parties. We consider both the synchronous and the asynchronous communication models. In the synchronous setting, our protocol is secure against a static malicious adversary corrupting less than a 1/31/31/3 fraction of the parties. In the asynchronous setting, we allow the adversary to corrupt less than a 1/81/81/8 fraction of parties. For any deterministic function that can be computed by an arithmetic circuit with mmm gates, both of our protocols require each party to send a number of field elements and perform an amount of computation that is O~(m/n+n)\tilde{O}(m/n + \sqrt n)O~(m/n+n​). We also show that our protocols provide perfect and universally-composable security. To achieve our asynchronous MPC result, we define the \emph{threshold counting problem} and present a distributed protocol to solve it in the asynchronous setting. This protocol is load balanced, with computation, communication and latency complexity of O(log⁡n)O(\log{n})O(logn), and can also be used for designing other load-balanced applications in the asynchronous communication model.