Constant-Round Asynchronous Multi-Party Computation Based on One-Way Functions

Constant-Round Asynchronous Multi-Party Computation Secure multi-party computation (MPC) allows several mutually distrustful parties to securely compute a joint function of their inputs and exists in two main variants: In *synchronous* MPC parties are connected by a synchronous network with a global clock, and protocols proceed in *rounds* with strong delivery guarantees, whereas *asynchronous* MPC protocols can be deployed even in networks that deliver messages in an arbitrary order and impose arbitrary delays on them. The two models---synchronous and asynchronous---have to a large extent developed in parallel with results on both feasibility and asymptotic efficiency improvements in either track. The most notable gap in this parallel development is with respect to round complexity. In particular, although under standard assumptions on a synchronous communication network (availability of secure channels and broadcast), synchronous MPC protocols with (exact) constant rounds have been constructed, to the best of our knowledge, thus far no constant-round asynchronous MPC protocols are known, with the best protocols requiring a number of rounds that is linear in the multiplicative depth of the arithmetic circuit computing the desired function. In this work we close this gap by providing the first constant-round asynchronous MPC protocol. Our protocol is optimally resilient (i.e., it tolerates up to $t<n/3$ corrupted parties), adaptively secure, and makes black-box use of a pseudo-random function. It works under the standard network assumptions for protocols in the asynchronous MPC setting, namely, a complete network of point-to-point (secure) asynchronous channels with eventual delivery and asynchronous Byzantine agreement (aka consensus). We provide formal definitions of these primitives and a proof of security in the Universal Composability framework

Asymmetric Multi-Party Computation

Current protocols for Multi-Party Computation (MPC) consider the setting where all parties have access to similar resources. For example, all parties have access to channels bounded by the same worst-case delay upper bound ?, and all channels have the same cost of communication. As a consequence, the overall protocol performance (resp. the communication cost) may be heavily affected by the slowest (resp. the most expensive) channel, even when most channels are fast (resp. cheap). Given the state of affairs, we initiate a systematic study of asymmetric MPC. In asymmetric MPC, the parties are divided into two categories: fast and slow parties, depending on whether they have access to high-end or low-end resources. We investigate two different models. In the first, we consider asymmetric communication delays: Fast parties are connected via channels with small delay ? among themselves, while channels connected to (at least) one slow party have a large delay ? ? ?. In the second model, we consider asymmetric communication costs: Fast parties benefit from channels with cheap communication, while channels connected to a slow party have an expensive communication. We provide a wide range of positive and negative results exploring the trade-offs between the achievable number of tolerated corruptions t and slow parties s, versus the round complexity and communication cost in each of the models. Among others, we achieve the following results. In the model with asymmetric communication delays, focusing on the information-theoretic (i-t) setting: - An i-t asymmetric MPC protocol with security with abort as long as t+s < n and t < n/2, in a constant number of slow rounds. - We show that achieving an i-t asymmetric MPC protocol for t+s = n and with number of slow rounds independent of the circuit size implies an i-t synchronous MPC protocol with round complexity independent of the circuit size, which is a major problem in the field of round-complexity of MPC. - We identify a new primitive, asymmetric broadcast, that allows to consistently distribute a value among the fast parties, and at a later time the same value to slow parties. We completely characterize the feasibility of asymmetric broadcast by showing that it is possible if and only if 2t + s < n. - An i-t asymmetric MPC protocol with guaranteed output delivery as long as t+s < n and t < n/2, in a number of slow rounds independent of the circuit size. In the model with asymmetric communication cost, we achieve an asymmetric MPC protocol for security with abort for t+s < n and t < n/2, based on one-way functions (OWF). The protocol communicates a number of bits over expensive channels that is independent of the circuit size. We conjecture that assuming OWF is needed and further provide a partial result in this direction

Studies on Verifiable Secret Sharing, Byzantine Agreement and Multiparty Computation

This dissertation deals with three most important as well as fundamental problems in secure distributed computing, namely Verifiable Secret Sharing (VSS), Byzantine Agreement (BA) and Multiparty Computation (MPC). VSS is a two phase protocol (Sharing and Reconstruction) carried out among $n$ parties in the presence of a centralized adversary who can corrupt up to $t$ parties. Informally, the goal of the VSS protocol is to share a secret $s$, among the $n$ parties during the sharing phase in a way that would later allow for a unique reconstruction of this secret in the reconstruction phase, while preserving the secrecy of $s$ until the reconstruction phase. VSS is used as a key tool in MPC, BA and many other secure distributed computing problems. It can take many different forms, depending on the underlying network (synchronous or asynchronous), the nature (passive or active) and computing power (bounded or unbounded) of the adversary, type of security (cryptographic or information theoretic) etc. We study VSS in information theoretic setting over both synchronous as well as asynchronous network, considering an active unbounded powerful adversary. Our main contributions for VSS are: \begin{itemize} \item In synchronous network, we carry out in-depth investigation on the round complexity of VSS by allowing a probability of error in computation and show that existing lower bounds for the round complexity of error-free VSS can be circumvented by introducing a negligible probability of error. \item We study the communication and round efficiency of VSS in synchronous network and present a robust VSS protocol that is simultaneously communication efficient and round efficient. In addition, our protocol is the best known communication and round efficient protocol in the literature. \item In asynchronous network, we study the communication complexity of VSS and propose a number of VSS protocols. Our protocols are highly communication efficient and show significant improvement over the existing protocols in terms of communication complexity. \end{itemize} The next problem that we deal with is Byzantine Agreement (BA). BA is considered as one of the most fundamental primitives for fault tolerant distributed computing and cryptographic protocols. BA among a set of $n$ parties, each having a private input value, allows them to reach agreement on a common value even if some of the malicious parties (at most $t$) try to prevent agreement among the parties. Similar to the case of VSS, several models for BA have been proposed during the last three decades, considering various aspects like the underlying network, the nature and computing power of adversary, type of security. One of these models is BA over asynchronous network which is considered to be more realistic network than synchronous in many occasions. Though important, research in BA in asynchronous network has received much less attention in comparison to the BA protocols in synchronous network. Even the existing protocols for asynchronous BA involve high communication complexity and in general are very inefficient in comparison to their synchronous counterparts. We focus on BA in information theoretic setting over asynchronous network tolerating an active adversary having unbounded computing power and mainly work towards the communication efficiency of the problem. Our contributions for BA are as follows: \begin{itemize} \item We propose communication efficient asynchronous BA protocols that show huge improvement over the existing protocols in the same setting. Our protocols for asynchronous BA use our VSS protocols in asynchronous network as their vital building blocks. \item We also construct a communication optimal asynchronous BA protocol for sufficiently long message size. Precisely, our asynchronous BA communicates O(\ell n) bits for \ell bit message, for sufficiently large \ell. \end{itemize} The studies on VSS and BA naturally lead one towards MPC problems. The MPC can model almost any known cryptographic application and uses VSS as well as BA as building blocks. MPC enables a set of $n$ mutually distrusting parties to compute some function of their private inputs, such that the privacy of the inputs of the honest parties is guaranteed (except for what can be derived from the function output) even in the presence of an adversary corrupting up to $t$ of the parties and making them misbehave arbitrarily. Much like VSS and BA, MPC can also be studied in various models. Here, we attempt to solve MPC in information theoretic setting over synchronous as well as asynchronous network, tolerating an active unbounded powerful adversary. As for MPC, our main contributions are: \begin{itemize} \item Using one of our synchronous VSS protocol, we design a synchronous MPC that minimizes the communication and round complexity simultaneously, where existing MPC protocols try to minimize one complexity measure at a time (i.e the existing protocols minimize either communication complexity or round complexity). \item We study the communication complexity of asynchronous MPC protocols and design a number of protocols for the same that show significant gain in communication complexity in comparison to the existing asynchronous MPC protocols. \item We also study a specific instance of MPC problem called Multiparty Set Intersection (MPSI) and provide protocols for the same. \end{itemize} In brief, our work in this thesis has made significant advancement in the state-of-the-art research on VSS, BA and MPC by presenting several inherent lower bounds and efficient/optimal solutions for the problems in terms of their key parameters such as communication complexity and time/round complexity. Thus our work has made a significant contribution to the field of secure distributed computing by carrying out a foundation research on the three most important problems of this field

Efficient Asynchronous Verifiable Secret Sharing and Multiparty Computation

Secure Multi-Party Computation (MPC) providing information theoretic security allows a set of n parties to securely compute an agreed function F over a finite field ${\mathbb F}$, even if t parties are under the control of a computationally unbounded active adversary. Asynchronous MPC (AMPC) is an important variant of MPC, which works over an asynchronous network. It is well known that perfect AMPC is possible if and only if n \geq 4t+1, while statistical AMPC is possible if and only if n \geq 3t+1. In this paper, we study the communication complexity of AMPC protocols (both statistical and perfect) designed with exactly n = 4t+1 parties. Our major contributions in this paper are as follows: 1. Asynchronous Verifiable Secret Sharing (AVSS) is one of the main building blocks for AMPC. In this paper, we design two AVSS protocols with 4t+1 parties: the first one is statistically secure and has non-optimal resilience, while the second one is perfectly secure and has optimal resilience. Both these schemes achieve a common interesting property, which was not achieved by the previous schemes. Specifically, our AVSS schemes allow to share a secret through a polynomial of degree at most d, where t \leq d \leq 2t. In contrast, the existing AVSS schemes can share a secret only through a polynomial of degree at most t. The new property of our AVSS simplifies the degree reduction step for the evaluation of multiplication gates in an AMPC protocol. 2.Using our statistical AVSS, we design a statistical AMPC protocol with n = 4t+1 which communicates O(n^2) field elements per multiplication gate. Though this protocol has non-optimal resilience, it significantly improves the communication complexity of the existing statistical AMPC protocols. 3. We then present a perfect AMPC protocol with n = 4t+1 (using our perfect AVSS scheme), which also communicates O(n^2) field elements per multiplication gate. This protocol improves on our statistical AMPC protocol as it has optimal resilience. To the best of our knowledge, this is the most communication efficient perfect AMPC protocol in the information theoretic setting

Optimally-resilient Unconditionally-secure Asynchronous Multi-party Computation Revisited

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 $t < \frac{n}{3}$ parties. Our protocol needs a communication of ${\cal O}(n^4)$ 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 ${\cal O}(n^5)$ 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

Asynchronous Multi-Party Quantum Computation

Multi-party quantum computation (MPQC) allows a set of parties to securely compute a quantum circuit over private quantum data. Current MPQC protocols rely on the fact that the network is synchronous, i.e., messages sent are guaranteed to be delivered within a known fixed delay upper bound, and unfortunately completely break down even when only a single message arrives late. Motivated by real-world networks, the seminal work of Ben-Or, Canetti and Goldreich (STOC\u2793) initiated the study of multi-party computation for classical circuits over asynchronous networks, where the network delay can be arbitrary. In this work, we begin the study of asynchronous multi-party quantum computation (AMPQC) protocols, where the circuit to compute is quantum. Our results completely characterize the optimal achievable corruption threshold: we present an n-party AMPQC protocol secure up to t < n/4 corruptions, and an impossibility result when t ? n/4 parties are corrupted. Remarkably, this characterization differs from the analogous classical setting, where the optimal corruption threshold is t < n/3

Asymmetric Multi-Party Computation

Current protocols for Multi-Party Computation (MPC) consider the setting where all parties have access to similar resources. For example, all parties have access to channels bounded by the same worst-case delay upper bound $\Delta$, and all channels have the same cost of communication. As a consequence, the overall protocol performance (resp. the communication cost) may be heavily affected by the slowest (resp. the most expensive) channel, even when most channels are fast (resp. cheap). Given the state of affairs, we initiate a systematic study of \u27asymmetric\u27 MPC. In asymmetric MPC, the parties are divided into two categories: fast and slow parties, depending on whether they have access to high-end or low-end resources. We investigate two different models. In the first, we consider asymmetric communication delays: Fast parties are connected via channels with small delay $\delta$ among themselves, while channels connected to (at least) one slow party have a large delay $\Delta \gg \delta$. In the second model, we consider asymmetric communication costs: Fast parties benefit from channels with cheap communication, while channels connected to a slow party have an expensive communication. We provide a wide range of positive and negative results exploring the trade-offs between the achievable number of tolerated corruptions $t$ and slow parties $s$, versus the round complexity and communication cost in each of the models. Among others, we achieve the following results. In the model with asymmetric communication delays, focusing on the information-theoretic (i-t) setting: - An i-t asymmetric MPC protocol with security with abort as long as $t+s < n$ and $t<n/2$, in a constant number of slow rounds. - We show that achieving an i-t asymmetric MPC protocol for $t+s = n$ and with number of slow rounds independent of the circuit size implies an i-t synchronous MPC protocol with round complexity independent of the circuit size, which is a major problem in the field of round-complexity of MPC. - We identify a new primitive, \emph{asymmetric broadcast}, that allows to consistently distribute a value among the fast parties, and at a later time the same value to slow parties. We completely characterize the feasibility of asymmetric broadcast by showing that it is possible if and only if $2t + s < n$. - An i-t asymmetric MPC protocol with guaranteed output delivery as long as $t+s < n$ and $t<n/2$, in a number of slow rounds independent of the circuit size. In the model with asymmetric communication cost, we achieve an asymmetric MPC protocol for security with abort for $t+s<n$ and $t<n/2$, based on one-way functions (OWF). The protocol communicates a number of bits over expensive channels that is independent of the circuit size. We conjecture that assuming OWF is needed and further provide a partial result in this direction

On the Communication Efficiency of Statistically-Secure Asynchronous MPC with Optimal Resilience

Secure multi-party computation (MPC) is a fundamental problem in secure distributed computing. An MPC protocol allows a set of $n$ mutually distrusting parties with private inputs to securely compute any publicly-known function of their inputs, by keeping their respective inputs as private as possible. While several works in the past have addressed the problem of designing communication-efficient MPC protocols in the synchronous communication setting, not much attention has been paid to the design of efficient MPC protocols in the asynchronous communication setting. In this work, we focus on the design of efficient asynchronous MPC (AMPC) protocol with statistical security, tolerating a computationally unbounded adversary, capable of corrupting up to $t$ parties out of the $n$ parties. The seminal work of Ben-Or, Kelmer and Rabin (PODC 1994) and later Abraham, Dolev and Stern (PODC 2020) showed that the optimal resilience for statistically-secure AMPC is $t < n/3$. Unfortunately, the communication complexity of the protocol presented by Ben-Or et al is significantly high, where the communication complexity per multiplication is $\Omega(n^{13} \kappa^2 \log n)$ bits (where $\kappa$ is the statistical-security parameter). To the best of our knowledge, no work has addressed the problem of improving the communication complexity of the protocol of Ben-Or at al. In this work, our main contributions are the following. -- We present a new statistically-secure AMPC protocol with the optimal resilience $t < n/3$ and where the communication complexity is ${\mathcal O}(n^4 \kappa)$ bits per multiplication. Apart from improving upon the communication complexity of the protocol of Ben-Or et al, our protocol is relatively simpler and based on very few sub-protocols, unlike the protocol of Ben-Or et al which involves several layers of subprotocols. A central component of our AMPC protocol is a new and simple protocol for verifiable asynchronous complete secret-sharing (ACSS), which is of independent interest. -- As a side result, we give the security proof for our AMPC protocol in the standard universal composability (UC) framework of Canetti (FOCS 2001, JACM 2020), which is now the defacto standard for proving the security of cryptographic protocols. This is unlike the protocol of Ben-Or et al, which was missing the formal security proofs

On Communication-Efficient Asynchronous MPC with Adaptive Security

Secure multi-party computation (MPC) allows a set of $n$ parties to jointly compute an arbitrary computation over their private inputs. Two main variants have been considered in the literature according to the underlying communication model. Synchronous MPC protocols proceed in rounds, and rely on the fact that the communication network provides strong delivery guarantees within each round. Asynchronous MPC protocols achieve security guarantees even when the network delay is arbitrary. While the problem of MPC has largely been studied in both variants with respect to both feasibility and efficiency results, there is still a substantial gap when it comes to communication complexity of adaptively secure protocols. Concretely, while adaptively secure synchronous MPC protocols with linear communication are known for a long time, the best asynchronous protocol communicates $\mathcal{O}(n^4 \kappa)$ bits per multiplication. In this paper, we make progress towards closing this gap by providing two protocols. First, we present an adaptively secure asynchronous protocol with optimal resilience $t<n/3$ and $\mathcal{O}(n^2 \kappa)$ bits of communication per multiplication, improving over the state of the art protocols in this setting by a quadratic factor in the number of parties. The protocol has cryptographic security and follows the CDN approach [Eurocrypt\u2701], based on additive threshold homomorphic encryption. Second, we show an optimization of the above protocol that tolerates up to $t<(1-\epsilon)n/3$ corruptions and communicates \mathcal{O}(n\cdot \poly(\kappa)) bits per multiplication under stronger assumptions

Byzantine Agreement Decomposed: Honest Majority Asynchronous Total-Order Broadcast from Reliable Broadcast

It is well-known that Asynchronous Total Order Broadcast (ATOB) requires randomisation and that at most $t < n/3$ out of $n$ players are corrupted. This is opposed to synchronous total-order broadcast (STOB) which can tolerate $t < n/2$ corruptions and can be deterministic. We show that these requirements can be conceptually separated, by constructing an ATOB protocol which tolerates $t < n/2$ corruptions from blackbox use of Common Coin and Reliable Broadcast. We show the power of this conceptually simple contribution by reproving, using simpler protocols, existing results on STOB with optimistic responsiveness and asynchronous fallback. We also use the framework to prove the first ATOB with sub-quadratic communication and optimal corruption threshold $t < n/3$, new ATOBs with covert security and mixed adversary structures, and a new STOB with asymmetric synchrony assumptions