In this work, we study the approximate consensus problem in asynchronous message-passing networks where some nodes may become Byzantine faulty. We answer an open problem raised by Tseng and Vaidya, 2012, proposing the first algorithm of optimal resilience for directed networks. Interestingly, our results show that the tight condition on the underlying communication networks for asynchronous Byzantine approximate consensus coincides with the tight condition for synchronous Byzantine exact consensus. Our results can be viewed as a non-trivial generalization of the algorithm by Abraham et al., 2004, which applies to the special case of complete networks. The tight condition and techniques identified in the paper shed light on the fundamental properties for solving approximate consensus in asynchronous directed networks.
We show that asynchronous $t$ faults Byzantine system is equivalent to asynchronous $t$-resilient system, where unbeknownst to all, the private inputs of at most $t$ processors were altered and installed by a malicious oracle. The immediate ramification is that dealing with asynchronous Byzantine systems does not call for new topological methods, as was recently employed by various researchers: Asynchronous Byzantine is a standard asynchronous system with an input caveat. It also shows that two recent independent investigations of vector $ε$-agreement in the Byzantine model, and then in the fa…
This report contains two related sets of results with different assumptions on synchrony. The first part is about iterative algorithms in synchronous systems. Following our previous work on synchronous iterative approximate Byzantine consensus (IABC) algorithms, we provide a more intuitive tight necessary and sufficient condition for the existence of such algorithms in synchronous networks1. We believe this condition and the previous results also hold in partially asynchronous algorithmic model. In the second part of the report, we explore the problem in asynchronous networks. While the tradit…
In this paper, we give a deterministic two-step Byzantine consensus protocol that achieves safety and liveness. A two-step Byzantine consensus protocol only needs two communication steps to commit in the absence of faults. Most two-step Byzantine consensus protocols exploit optimism and require a recovery protocol in the presence of faults. In this paper, we give a simple two-step Byzantine consensus protocol that does not need a recovery protocol.
We study the Byzantine lattice agreement (BLA) problem in asynchronous distributed message passing systems. In the BLA problem, each process proposes a value from a join semi-lattice and needs to output a value also in the lattice such that all output values of correct processes lie on a chain despite the presence of Byzantine processes. We present an algorithm for this problem with round complexity of $O(\log f)$ which tolerates $f < \frac{n}{5}$ Byzantine failures in the asynchronous setting without digital signatures, where $n$ is the number of processes. We also show how this algorithm can…
The Byzantine agreement problem is considered to be a core problem in distributed systems. For example, Byzantine agreement is needed to build a blockchain, a totally ordered log of records. Blockchains are asynchronous distributed systems, fault-tolerant against Byzantine nodes. In the literature, the asynchronous byzantine agreement problem is studied in a fully connected network model where every node can directly send messages to every other node. This assumption is questionable in many real-world environments. In the reality, nodes might need to communicate by means of an incomplete netwo…
In this paper, we explore the problem of iterative approximate Byzantine consensus in arbitrary directed graphs. In particular, we prove a necessary and sufficient condition for the existence of iterative byzantine consensus algorithms. Additionally, we use our sufficient condition to examine whether such algorithms exist for some specific graphs.
In this report, we investigate the multi-valued Byzantine consensus problem. We introduce two algorithms: the first one achieves traditional validity requirement for consensus, and the second one achieves a stronger "q-validity" requirement. Both algorithms are more efficient than the ones introduces in our recent PODC 2011 paper titled "Error-Free Multi-Valued Consensus with Byzantine Failures".
In this work, we explore iterative approximate Byzantine consensus algorithms that do not make explicit use of the global parameter of the graph, i.e., the upper-bound on the number of faults, f.
In this work we look at Byzantine consensus in asynchronous systems under the local broadcast model. In the local broadcast model, a message sent by any node is received identically by all of its neighbors in the communication network, preventing a faulty node from transmitting conflicting information to different neighbors. Our recent work has shown that in the synchronous setting, network connectivity requirements for Byzantine consensus are lower under the local broadcast model as compared to the classical point-to-point communication model. Here we show that the same is not true in the asy…
In this work, we extend the topology-based approach for characterizing computability in asynchronous crash-failure distributed systems to asynchronous Byzantine systems. We give the first theorem with necessary and sufficient conditions to solve arbitrary tasks in asynchronous Byzantine systems where an adversary chooses faulty processes. In our adversarial formulation, outputs of non-faulty processes are constrained in terms of inputs of non-faulty processes only. For colorless tasks, an important subclass of distributed problems, the general result reduces to an elegant model that effectivel…
We provide a new protocol for Validated Asynchronous Byzantine Agreement. Validated (multi-valued) Asynchronous Byzantine Agreement is a key building block in constructing Atomic Broadcast and fault-tolerant state machine replication in the asynchronous setting. Our protocol can withstand the optimal number $f<n/3$ of Byzantine failures and reaches agreement in the asymptotically optimal expected $O(1)$ running time. Honest parties in our protocol send only an expected $O(n^2)$ messages where each message contains a value and a constant number of signatures. Hence our total expected communicat…
King and Saia were the first to break the quadratic word complexity bound for Byzantine Agreement in synchronous systems against an adaptive adversary, and Algorand broke this bound with near-optimal resilience (first in the synchronous model and then with eventual-synchrony). Yet the question of asynchronous sub-quadratic Byzantine Agreement remained open. To the best of our knowledge, we are the first to answer this question in the affirmative. A key component of our solution is a shared coin algorithm based on a VRF. A second essential ingredient is VRF-based committee sampling, which we fo…
This paper presents a simple and efficient reliable broadcast algorithm for asynchronous message-passing systems made up of $n$ processes, among which up to $t<n/5$ may behave arbitrarily (Byzantine processes). This algorithm requires two communication steps and $n^2-1$ messages. When compared to Bracha's algorithm, which is resilience optimal ($t<n/3$) and requires three communication steps and $2n^2-n-1$ messages, the proposed algorithm shows an interesting tradeoff between communication efficiency and $t$-resilience.
It is a critical matter for a blockchain system whether a Byzantine fault tolerance (BFT) can be guaranteed during a consensus process. Can connected vehicles (CVs) achieve the BFT consensus when the vehicles keep mobile? This paper seeks an answer to this fundamental question. It focuses on characterizing the impact of mobility on the performance of a BFT consensus among CVs.
A stabilizing Byzantine single-writer single-reader (SWSR) regular register, which stabilizes after the first invoked write operation, is first presented. Then, new/old ordering inversions are eliminated by the use of a (bounded) sequence number for writes, obtaining a practically stabilizing SWSR atomic register. A practically stabilizing Byzantine single-writer multi-reader (SWMR) atomic register is then obtained by using several copies of SWSR atomic registers. Finally, bounded time-stamps, with a time-stamp per writer, together with SWMR atomic registers, are used to construct a practicall…
This work addresses Byzantine vector consensus (BVC), wherein the input at each process is a d-dimensional vector of reals, and each process is expected to decide on a decision vector that is in the convex hull of the input vectors at the fault-free processes [3, 8]. The input vector at each process may also be viewed as a point in the d-dimensional Euclidean space R^d, where d > 0 is a finite integer. Recent work [3, 8] has addressed Byzantine vector consensus in systems that can be modeled by a complete graph. This paper considers Byzantine vector consensus in incomplete graphs. In particula…
Given a system with $n > 3t + 1$ processes, where $t$ is the tolerated number of faulty ones, we present a fast asynchronous Byzantine agreement protocol that can reach agreement in $O(t)$ expected running time. This improves the $O(n^2)$ expected running time of Abraham, Dolev, and Halpern [PODC 2008]. Furthermore, if $n = (3 + \varepsilon) t$ for any $\varepsilon > 0$, our protocol can reach agreement in $O (1 / \varepsilon)$ expected running time. This improves the result of Feldman and Micali [STOC 1988] (with constant expected running time when $n > 4 t$).
It is a common belief that Byzantine fault-tolerant solutions for consensus are significantly slower than their crash fault-tolerant counterparts. Indeed, in PBFT, the most widely known Byzantine fault-tolerant consensus protocol, it takes three message delays to decide a value, in contrast with just two in Paxos. This motivates the search for fast Byzantine consensus algorithms that can produce decisions after just two message delays \emph{in the common case}, e.g., under the assumption that the current leader is correct and not suspected by correct processes. The (optimal) two-step latency c…
This article presents a signature-free distributed algorithm which builds an atomic read/write shared memory on top of an $n$-process asynchronous message-passing system in which up to $t<n/3$ processes may commit Byzantine failures. From a conceptual point of view, this algorithm is designed to be as close as possible to the algorithm proposed by Attiya, Bar-Noy and Dolev (JACM 1995), which builds an atomic register in an $n$-process asynchronous message-passing system where up to $t<n/2$ processes may crash. The proposed algorithm is particularly simple. It does not use cryptography to cope …
This paper considers the problem of reliable broadcast in asynchronous authenticated systems, in which n processes communicate using signed messages and up to t processes may behave arbitrarily (Byzantine processes). In addition, for each message m broadcast by a correct (i.e., non-Byzantine) process, a message adversary may prevent up to d correct processes from receiving m. (This message adversary captures network failures such as transient disconnections, silent churn, or message losses.) Considering such a "double" adversarial context and assuming n > 3t + 2d, a reliable broadcast algorith…
This paper considers the problem of achieving exact Byzantine consensus in a synchronous system under a local-broadcast communication model. The nodes communicate with each other via message-passing. The communication network is modeled as an undirected graph, with each vertex representing a node in the system. Under the local-broadcast communication model, when any node transmits a message, all its neighbors in the communication graph receive the message reliably. This communication model is motivated by wireless networks. In this work, we present necessary and sufficient conditions on the un…
Approximate byzantine consensus is a fundamental problem of distributed computing. This paper presents a novel algorithm for approximate byzantine consensus, called Relay-ABC. The algorithm allows machines to achieve approximate consensus to arbitrary exactness in the presence of byzantine failures. The algorithm relies on the usage of a relayed messaging system and signed messages with unforgeable signatures that are unique to each node. The use of signatures and relays allows the strict necessary network conditions of traditional approximate byzantine consensus algorithms to be circumvented.…
Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantin…
To circumvent the FLP impossibility result in a deterministic way several protocols have been proposed on top of an asynchronous distributed system enriched with additional assumptions. In the context of Byzantine failures for systems where at most t processes may exhibit a Byzantine behavior, two approaches have been investigated to solve the consensus problem.The first, relies on the addition of synchrony, called Timer-Based, but the second is based on the pattern of the messages that are exchanged, called Time-Free. This paper shows that both types of assumptions are not antagonist and can …
Iterative Approximate Byzantine Consensus (IABC) is a fundamental problem of fault-tolerant distributed computing where machines seek to achieve approximate consensus to arbitrary exactness in the presence of Byzantine failures. We present a novel algorithm for this problem, named Relay-IABC, which relies on the usage of a multi-hop relayed messaging system and crytographically secure message signatures. The use of signatures and relays allows the strict necessary network conditions of traditional IABC algorithms to be circumvented. In addition, we show evidence that Relay-IABC achieves faster…
Consider a synchronous point-to-point network of n nodes connected by directed links, wherein each node has a binary input. This paper proves a tight necessary and sufficient condition on the underlying communication topology for achieving Byzantine consensus among these nodes in the presence of up to f Byzantine faults. We derive a necessary condition, and then we provide a constructive proof of sufficiency by presenting a Byzantine consensus algorithm for directed graphs that satisfy the necessary condition. Prior work has developed analogous necessary and sufficient conditions for undirecte…
Replicated services are inherently vulnerable to failures and security breaches. In a long-running system, it is, therefore, indispensable to maintain a reconfiguration mechanism that would replace faulty replicas with correct ones. An important challenge is to enable reconfiguration without affecting the availability and consistency of the replicated data: the clients should be able to get correct service even when the set of service replicas is being updated. In this paper, we address the problem of reconfiguration in the presence of Byzantine failures: faulty replicas or clients may arbitra…
One of the most celebrated problems of fault-tolerant distributed computing is the consensus problem. It was shown to abstract a myriad of problems in which processes have to agree on a single value. Consensus applications include fundamental services for the environments of the Cloud or Blockchain. In such challenging environments, malicious behavior is often modeled as adversarial Byzantine faults. At OPODIS 2010, Moste}faoui and Raynal, in short, MR, presented a Byzantine- and intrusion-tolerant solution to consensus in which the decided value cannot be a value proposed only by Byzantine pr…
This paper describes a simple and efficient asynchronous Binary Byzantine faulty tolerant consensus algorithm. In the algorithm, non-faulty nodes perform an initial broadcast followed by a executing a series of rounds each consisting of a single message broadcast plus the computation of a global random coin using threshold signatures. Each message is accompanied by a cryptographic proof of its validity. Up to one third of the nodes can be faulty and termination is expected in a constant number of rounds. An optimization is described allowing the round message plus the coin message to be combin…
In this paper we propose Aleph, a leaderless, fully asynchronous, Byzantine fault tolerant consensus protocol for ordering messages exchanged among processes. It is based on a distributed construction of a partially ordered set and the algorithm for reaching a consensus on its extension to a total order. To achieve the consensus, the processes perform computations based only on a local copy of the data structure, however, they are bound to end with the same results. Our algorithm uses a dual-threshold coin-tossing scheme as a randomization strategy and establishes the agreement in an expected …
Consider an asynchronous network in a shared-memory environment consisting of n nodes. Assume that up to f of the nodes might be Byzantine (n > 12f), where the adversary is full-information and dynamic (sometimes called adaptive). In addition, the non-Byzantine nodes may undergo transient failures. Nodes advance in atomic steps, which consist of reading all registers, performing some calculation and writing to all registers. This paper contains three main contributions. First, the clock-function problem is defined, which is a generalization of the clock synchronization problem. This generaliza…
This paper explores the problem of reaching approximate consensus in synchronous point-to-point networks, where each directed link of the underlying communication graph represents a communication channel between a pair of nodes. We adopt the transient Byzantine link failure model [15, 16], where an omniscient adversary controls a subset of the directed communication links, but the nodes are assumed to be fault-free. Recent work has addressed the problem of reaching approximate consen- sus in incomplete graphs with Byzantine nodes using a restricted class of iterative algorithms that maintain o…
The Dolev-Reischuk bound says that any deterministic Byzantine consensus protocol has (at least) quadratic communication complexity in the worst case. While it has been shown that the bound is tight in synchronous environments, it is still unknown whether a consensus protocol with quadratic communication complexity can be obtained in partial synchrony. Until now, the most efficient known solutions for Byzantine consensus in partially synchronous settings had cubic communication complexity (e.g., HotStuff, binary DBFT). This paper closes the existing gap by introducing SQuad, a partially synchr…
We explore asynchronous unison in the presence of systemic transient and permanent Byzantine faults in shared memory. We observe that the problem is not solvable under less than strongly fair scheduler or for system topologies with maximum node degree greater than two. We present a self-stabilizing Byzantine-tolerant solution to asynchronous unison for chain and ring topologies. Our algorithm has minimum possible containment radius and optimal stabilization time.
Consider an asynchronous system with private channels and $n$ processes, up to $t$ of which may be faulty. We settle a longstanding open question by providing a Byzantine agreement protocol that simultaneously achieves three properties: 1. (optimal) resilience: it works as long as $n>3t$ 2. (almost-sure) termination: with probability one, all nonfaulty processes terminate 3. (polynomial) efficiency: the expected computation time, memory consumption, message size, and number of messages sent are all polynomial in $n$. Earlier protocols have achieved only two of these three properties. In partic…
Partially synchronous Byzantine consensus protocols typically structure their execution into a sequence of views, each with a designated leader process. The key to guaranteeing liveness in these protocols is to ensure that all correct processes eventually overlap in a view with a correct leader for long enough to reach a decision. We propose a simple view synchronizer abstraction that encapsulates the corresponding functionality for Byzantine consensus protocols, thus simplifying their design. We present a formal specification of a view synchronizer and its implementation under partial synchro…
Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [3, 7]. Recent work has yielded approximate Byzantine consensus algorithms for the case when the input at each node is a d-dimensional vector, and the nodes must reach consensus on a vector in the convex hull of the input vectors at the fault-free nodes [8, 12]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [8, 12] require the fault-free nodes to decide on a point in the d-dimensional space. In this paper, we…
We consider Byzantine consensus in a synchronous system where nodes are connected by a network modeled as a directed graph, i.e., communication links between neighboring nodes are not necessarily bi-directional. The directed graph model is motivated by wireless networks wherein asymmetric communication links can occur. In the classical point-to-point communication model, a message sent on a communication link is private between the two nodes on the link. This allows a Byzantine faulty node to equivocate, i.e., send inconsistent information to its neighbors. This paper considers the local broad…
Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [4, 8]. Recent work has yielded approximate Byzantine consensus algorithms for the case when the input at each node is a d-dimensional vector, and the nodes must reach consensus on a vector in the convex hull of the input vectors at the fault-free nodes [9, 13]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [9, 13] require the fault-free nodes to decide on a point in the d-dimensional space. In our recent wor…
Consider an asynchronous system where each node begins with some point in $\mathbb{R}^m$. Given some fixed $ε> 0$, we wish to have every nonfaulty node eventually output a point in $\mathbb{R}^m$, where all outputs are within distance $ε$ of each other, and are within the convex hull of the original nonfaulty inputs. This problem, when some of the nodes are adversarial, is known as the ``Byzantine Asynchronous Multidimensional Approximate Agreement'' problem. Previous landmark work by Mendes et al. and Vaidya et al. presented two solutions to the problem. Both of these solutions require expone…
Modular methods to transform Byzantine consensus protocols into ones that are fast and communication efficient in the common cases are presented. Small and short protocol segments called layers are custom designed to optimize performance in the common case. When composed with a Byzantine consensus protocol of choice, they allow considerable control over the tradeoff in the combined protocol's behavior in the presence of failures and its performance in their absence. When runs are failure free in the common case, the resulting protocols decide in two rounds and require $2nt$ bits of communicati…
Existing Byzantine fault-tolerant (BFT) consensus protocols address only threshold failures, where the participating nodes fail independently of each other, each one fails equally likely, and the protocol's guarantees follow from a simple bound on the number of faulty nodes. With the widespread deployment of Byzantine consensus in blockchains and distributed ledgers today, however, more sophisticated trust assumptions are needed. This paper presents the first implementation of BFT consensus with generalized quorums. It starts from a number of generalized trust structures motivated by practice …
In this work we consider Byzantine Consensus on undirected communication graphs under the local broadcast model. In the classical point-to-point communication model the messages exchanged between two nodes $u, v$ on an edge $uv$ of $G$ are private. This allows a faulty node to send conflicting information to its different neighbours, a property called equivocation. In contrast, in the local broadcast communication model considered here, a message sent by node $u$ is received identically by all of its neighbours. This restriction to broadcast messages provides non-equivocation even for faulty n…
In this work, we consider a generalized fault model that can be used to represent a wide range of failure scenarios, including correlated failures and non-uniform node reliabilities. This fault model is general in the sense that fault models studied in prior related work, such as f -total and f -local models, are special cases of the generalized fault model. Under the generalized fault model, we explore iterative approximate Byzantine consensus (IABC) algorithms in arbitrary directed networks. We prove a necessary and sufficient condition for the existence of IABC algorithms. The use of the ge…
This short paper presents a necessary condition for Byzantine $k$-set agreement in (synchronous or asynchronous) message-passing systems and asynchronous shared memory systems where the processes communicate through atomic single-writer multi-reader registers. It gives a proof, which is particularly simple, that $k$-set agreement cannot be solved $t$-resiliently in an $n$-process system when $n \leq 2t + \frac{t}{k}$. This bound is tight for the case $k=1$ (Byzantine consensus) in synchronous message-passing systems.
This work performs an experimental evaluation of four asynchronous binary Byzantine consensus algorithms [11,16,18] in various configurations. In addition to being asynchronous these algorithms run in rounds, tolerate up to one third of faulty nodes, use $O(n^2)$ messages, and use randomized common coins to terminate in an expected constant number of rounds. Each of the four algorithms have different requirements for the random coin, for the number of messages needed per round, whether or not cryptographic signatures are needed, among other details. Two different non-interactive threshold comm…
Traditional Byzantine Fault Tolerance (BFT) state machine replication protocols assume a partial synchrony model, leading to a design where a leader replica drives the protocol and is replaced after a timeout. Recently, we witnessed a surge of asynchronous BFT protocols that use randomization to remove the assumptions of bounds on message delivery times, making them more resilient to adverse network conditions. However, these protocols still fall short of being practical across a broad range of scenarios due to their cubic communication costs, use of expensive primitives, and overall protocol …
The BG-simulation is a powerful reduction algorithm designed for asynchronous read/write crash-prone systems. It allows a set of $(t+1)$ asynchronous sequential processes to wait-free simulate (i.e., despite the crash of up to $t$ of them) an arbitrary number $n$ of processes under the assumption that at most $t$ of them may crash. The BG simulation shows that, in read/write systems, the crucial parameter is not the number $n$ of processes, but the upper bound $t$ on the number of process crashes. The paper extends the concept of BG simulation to asynchronous message-passing systems prone to B…
Université Pierre et Marie Curie - Paris 6（巴黎第六大学皮埃尔与玛丽·居里大学）；LIP6-CNRS（巴黎第六大学计算机实验室（法国国家科学研究中心））
We propose the first deterministic algorithm that tolerates up to $f$ byzantine faults in $3f+1$-sized networks and performs in the asynchronous CORDA model. Our solution matches the previously established lower bound for the semi-synchronous ATOM model on the number of tolerated Byzantine robots. Our algorithm works under bounded scheduling assumptions for oblivious robots moving in a uni-dimensional space.
Byzantine fault tolerance (BFT) consensus is a fundamental primitive for distributed computation. However, BFT protocols suffer from the ordering manipulation, in which an adversary can make front-running. Several protocols are proposed to resolve the manipulation problem, but there are some limitations for them. The batch-based protocols such as Themis has significant performance loss because of the use of complex algorithms to find strongly connected components (SCCs). The timestamp-based protocols such as Pompe have simplified the ordering phase, but they are limited on fairness that the ad…
