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IEEE Computer Security Foundations Workshop (CSFW)

Unconditionally Secure MPC for Boolean Circuits With Constant Online Communication

Zhenkai Hu, Kang Yang, Yu Yu

Published Jul 8, 2024• 2 citations
Publisher

Abstract

Through tremendous efforts, the communication cost of secure multi-party computation (MPC) in the honest-majority setting has been significantly improved. In particular, the state-of-the-art honest-majority MPC protocol by Escudero et al. (CCS'22) takes 12 field elements in total per multiplication gate for arithmetic circuits in the online phase. However, it still requires$12 log (5n/4$) bits of online communication per AND gate for Boolean circuits. That is, for Boolean circuits, no MPC protocol with constant online communication is known. In this paper, we present an unconditionally secure MPC protocol for Boolean circuits in the honest-majority setting, which has constant online communication complexity and the offline communication complexity linear to the number$n$of parties. We first describe the semi-honest MPC protocol and then show how to extend it to achieve malicious security, where the maliciously secure protocol has the same communication cost as the semi-honest protocol. In particular, our protocol achieves the amortized communication cost 36 bits per AND gate in the online phase and 30n + 24 bits per AND gate in the offline phase.

DOI
10.1109/csf61375.2024.00028

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Through tremendous efforts, the communication cost of secure multi-party computation (MPC) in the honest-majority setting has been significantly improved. In particular, the state-of-the-art honest-majority MPC protocol by Escudero et al. (CCS'22) takes 12 field elements in total per multiplication gate for arithmetic circuits in the online phase. However, it still requires12log(5n/412 log (5n/412log(5n/4) bits of online communication per AND gate for Boolean circuits. That is, for Boolean circuits, no MPC protocol with constant online communication is known. In this paper, we present an unconditionally secure MPC protocol for Boolean circuits in the honest-majority setting, which has constant online communication complexity and the offline communication complexity linear to the numbernnnof parties. We first describe the semi-honest MPC protocol and then show how to extend it to achieve malicious security, where the maliciously secure protocol has the same communication cost as the semi-honest protocol. In particular, our protocol achieves the amortized communication cost 36 bits per AND gate in the online phase and 30n + 24 bits per AND gate in the offline phase.
Secure multiparty computation (MPC) enables a set of parties to securely carry out a joint computation of their private inputs without revealing anything but the output. Protocols for semi-honest adversaries guarantee security as long as the corrupted parties run the specified protocol and ensure that nothing is leaked in the transcript. In contrast, protocols for malicious adversaries guarantee security in the presence of arbitrary adversaries who can run any attack strategy. Security for malicious adversaries is typically what is needed in practice (and is always preferred), but comes at a significant cost. In this paper, we present the first protocol for a two-thirds honest majority that achieves security in the presence of malicious adversariesat essentially the exact same cost as the best known protocols for semi-honest adversaries. Our construction is not a general transformation and thus it is possible that better semi-honest protocols will be constructed which do not support our transformation. Nevertheless, for the current state-of-the-art for many parties (based on Shamir sharing), our protocol invokes the best semi-honest multiplication protocol exactly once per multiplication gate (plus some additional local computation that is negligible to the overall cost). Concretely, the best version of our protocol requires each party to send on average of just 2 2/3 elements per multiplication gate (when the number of multiplication gates is at least the number of parties). This is four times faster than the previous-best protocol of Barak et al. (ACM CCS 2018) for small fields, and twice as fast as the previous-best protocol of Chida et al. (CRYPTO 2018) for large fields.
Secure multiparty computation enables a set of parties to securely carry out a joint computation on their private inputs without revealing anything but the output. A particularly motivated setting is that of three parties with a single corruption (hereafter denoted 3PC). This 3PC setting is particularly appealing for two main reasons: (1) it admits more efficient MPC protocols than in other standard settings; (2) it allows in principle to achieve full security (and fairness). Highly efficient protocols exist within this setting with security against a semi-honest</> adversary; however, a significant gap remains between these and protocols with stronger security against a malicious</> adversary. In this paper, we narrow this gap within concretely efficient protocols. More explicitly, we have the following contributions: Concretely Efficient Malicious 3PC. We present an optimized 3PC protocol for arithmetic circuits over rings with (amortized) communication of 1 ring element per multiplication gate per party, matching the best semi-honest protocols. The protocol applies also to Boolean circuits, significantly improving over previous protocols even for small circuits. Our protocol builds on recent techniques of Boneh et al. (Crypto 2019) for sublinear zero-knowledge proofs on distributed data, together with an efficient semi-honest protocol based on replicated secret sharing (Araki et al., CCS 2016). We present a concrete analysis of communication and computation costs, including several optimizations. For example, for 40-bit statistical security, and Boolean circuit with a million (nonlinear) gates, the overhead on top of the semi-honest protocol can involve less than 0.5KB of communication for the entire circuit,</> while the computational overhead is dominated by roughly 30 multiplications per gate in the field F247. In addition, we implemented and benchmarked the protocol for varied circuit sizes. Full Security. We augment the 3PC protocol to further provide full security</> (with guaranteed output delivery) while maintaining amortized 1 ring element communication per party per multiplication gate, and with hardly any impact on concrete efficiency. This is contrasted with the best previous 3PC protocols from the literature, which allow a corrupt party to mount a denial-of-service attack without being detected.
parties to securely compute an agreed function f over some finite field in the presence of a computationally unbounded adversary, who can maliciously corrupt any t out of the n parties. Most of the known efficient MPC protocols are designed in the offline- online framework introduced in a seminal work by Beaver in CRYPTO 1991. In this framework, the parties generate shared random and private multiplication-triples during the offline phase, which are used later in the online phase for securely evaluating the multiplication gates in the circuit representing f . The efficiency of the MPC protocols in this framework then relies on efficient ways of implementing the offline phase. In this paper, we propose a new and simple framework for generating shared and private random multiplication triples with unconditional security. The existing protocols approach this problem by first producing shared pairs of private and random values, followed by securely computing the shared product of each pair of values. The latter task involves a multiplication protocol for shared values that are typically communication intensive. Our framework takes a completely different approach and shuns the use of multiplication protocol. Namely, we ask the parties to verifiably share random multiplication triples and then securely extract shared random multiplication triples unknown to the adversary, from the shared triples. Realizing our framework in the asynchronous and hybrid network setting,1 we present the first ever MPC protocols with a linear (in the number of parties) communication overhead per multiplication gate in the circuit representing f . These are significant improvements over the best known existing MPC protocols in the asynchronous and hybrid network setting with communication complexity O(n2) and O(n3), respectively. Our framework when applied to the synchronous setting results in round-efficient MPC protocols.
Secure multiparty computation (MPC) over Z2kis more efficient than computations over fields, and studying MPC protocols under malicious security has practical application value. Malicious security with a dishonest majority over rings remains challenging. The most popular approach is SPDZ2k, however, this is a specific protocol that does not support the transformation of any existing semi-honest MPC protocols into malicious security protocols. The zero knowledge proof (ZKP)-based compiler satisfies this requirement. Existing state-of-the-art protocols have logarithmic online communication overhead in terms of the circuit size |C|, and their direct application to rings is nontrivial as they were originally designed for finite fields. In this work, we investigate the communication overhead to develop malicious security protocols. We bridge the gap between malicious security with abort and semi-honest security, by constructing a “GMW-style” verification protocol to achieve malicious security in a dishonest majority setting. This approach incurs a constant online communication overhead by enhancing the machinery of zero-knowledge fully linear interactive oracle proof (zk-FLIOP). Additionally, we extend the zk-FLIOP to work over any ring by invoking reverse multiplication friendly embeddings (RMFEs). Our results show that the online communication complexity of the verification process depends on only the security parameter, the number of parties, and the ring size. Furthermore, for small-scale circuits over Z2, we designed a distributed lookup table argument where both the total communication complexity and the computational cost are independent of the circuit size but of the input wires.
We present a new approach to designing concretely efficient MPC protocols with semi-honest security in the dishonest majority setting. Motivated by the fact that within the dishonest majority setting the efficiency of most practical protocols does not depend on the number of honest parties, we investigate how to construct protocols which improve in efficiency as the number of honest parties increases. Our central idea is to take a protocol which is secure for n−1n-1n−1 corruptions and modify it to use short symmetric keys, with the aim of basing security on the concatenation of all honest parties’ keys. This results in a more efficient protocol tolerating fewer corruptions, whilst also introducing an LPN-style syndrome decoding assumption. We first apply this technique to a modified version of the semi-honest GMW protocol, using OT extension with short keys, to improve the efficiency of standard GMW with fewer corruptions. We also obtain more efficient constant-round MPC, using BMR-style garbled circuits with short keys, and present an implementation of the online phase of this protocol. Our techniques start to improve upon existing protocols when there are around n=20n=20n=20 parties with h=6h=6h=6 honest parties, and as these increase we obtain up to a 13 times reduction (for n=400,h=120n=400, h=120n=400,h=120 ) in communication complexity for our GMW variant, compared with the best-known GMW-based protocol modified to use the same threshold.
We put forth a new paradigm for secure multi-party computation (MPC) in the preprocessing model, where a feasible one-time setup can enable a lifetime of efficient online secure computations. Our protocols match the security guarantees and low costs of the cheapest category of MPC solutions, namely 3-party protocols (3PC) secure against a single malicious party, with the qualitative advantages that one party communicates data sublinear in the circuit size, and can go offline after its initial messages. This “2+ 1“-party structure can alternatively be instantiated between 2 parties with the aid of an (untrusted) dealer. Within such existing protocols, we provide comparable online performance while improving the storage and offline dealer-to-party communication requirements by more than 3 orders of magnitude. At the technical level, we build on the Fully Linear Interactive Oracle Proof (FLIOP)-based protocol design of Boyle et al. (CRYPTO 2021). We provide an extensive assortment of algorithmic and implementation-level optimizations, design efficient distributed proofs of well-formedness of complex FLIOP correlations, and make them circuit-independent. We implement and benchmark our end-to-end system against the state of the art in the 2+1 regime, a dealer-aided variant of SPDZ for Boolean circuits. We additionally extend our techniques to the(n+1)(n+1)(n+1)party setting, where a dealer aids general dishonest-majority MPC, and provide a variant of the protocol which further achieves security with “identifiable abort.”