Consider a distributed system with $n$ processors out of which $f$ can be Byzantine faulty. In the approximate agreement task, each processor $i$ receives an input value $x_i$ and has to decide on an output value $y_i$ such that
- the output values are in the convex hull of the non-faulty processors' input values,
- the output values are within distance $d$ of each other.
Classically, the values are assumed to be from an $m$-dimensional Euclidean space, where $m \ge 1$.
In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph $G$ and the goal is to output vertices that are within distance $d$ of each other in $G$, but still remain in the graph-induced convex hull of the input values. For $d=0$, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any $d \ge 1$, we show that the task is solvable in asynchronous systems when $G$ is chordal and $n > (ω+1)f$, where $ω$ is the clique number of~$G$. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.
For reaching efficient deterministic synchronous Byzantine agreement upon partially connected networks, the traditional broadcast primitive is extended and integrated with a general framework. With this, the Byzantine agreement is extended to fully connected bipartite networks and some bipartite bounded-degree networks. The complexity of the Byzantine agreement is lowered and optimized with the so-called Byzantine-levers under a general system structure. Some bipartite simulation of the butterfly networks and some finer properties of bipartite bounded-degree networks are also provided for buil…
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 recent years, Byzantine Agreement is being considered in increasing scales due to the proliferation of blockchains and other decentralized financial technologies. Consequently, a number of works have improved its communication complexity in various network models. In this short paper we survey recent advances and outline some open research questions on the subject.
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.
In this paper, we show that the protocol complex of a Byzantine synchronous system can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds, where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$. This topological property implies that $\lceil t/k \rceil + 1$ rounds are necessary to solve $k$-set agreement in Byzantine synchronous systems, compared to $\lfloor t/k \rfloor + 1$ rounds in synchronous crash-failure systems. We also show that our connectivity bound is tight as we indicate solutions to Byzantine $k$-set agreement in exactly $\lceil t/k \rceil + 1$ sy…
This is a correction by the authors to "Byzantine Agreement in Expected Polynomial Time" which appeared in the Journal of the ACM in 2016. It corrects a failure in the paper to consider the adversary's ability to decide the number of fair coinflips in an iteration, where this number ranges between n(n-t) and n(n-2t).
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.
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…
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.
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…
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…
Byzantine Agreement introduced in [Pease, Shostak, Lamport, 80] is a widely used building block of reliable distributed protocols. It simulates broadcast despite the presence of faulty parties within the network, traditionally using only private unicast links. Under such conditions, Byzantine Agreement requires more than 2/3 of the parties to be compliant. [Fitzi, Maurer, 00], constructed a Byzantine Agreement protocol for any compliant majority based on an additional primitive allowing transmission to any two parties simultaneously. They proposed a problem of generalizing these results to wid…
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 $n > 4 t$).
We provide the first protocol that solves Byzantine agreement with optimal early stopping ($\min\{f+2,t+1\}$ rounds) and optimal resilience ($n>3t$) using polynomial message size and computation. All previous approaches obtained sub-optimal results and used resolve rules that looked only at the immediate children in the EIG (\emph{Exponential Information Gathering}) tree. At the heart of our solution are new resolve rules that look at multiple layers of the EIG tree.
A set of mutually distrusting participants that want to agree on a common opinion must solve an instance of a Byzantine agreement problem. These problems have been extensively studied in the literature. However, most of the existing solutions assume that the participants are aware of $n$ -- the total number of participants in the system -- and $f$ -- an upper bound on the number of Byzantine participants. In this paper, we show that most of the fundamental agreement problems can be solved without affecting resiliency even if the participants do not know the values of (possibly changing) $n$ an…
In the Byzantine agreement problem, n nodes with possibly different input values aim to reach agreement on a common value in the presence of t < n/3 Byzantine nodes which represent arbitrary failures in the system. This paper introduces a generalization of Byzantine agreement, where the input values of the nodes are preference rankings of three or more candidates. We show that consensus on preferences, which is an important question in social choice theory, complements already known results from Byzantine agreement. In addition preferential voting raises new questions about how to approximate …
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 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., 20…
Permissionless blockchain protocols are known to consume an outrageous amount of computing power and suffer from a trade-off between latency and confidence in transaction confirmation. The recently proposed Algorand blockchain protocol employs Byzantine agreements and has shown transaction confirmation latency on the order of seconds. Its strong resilience to Denial-of-Service and Sybil attacks and its low computing power footprint make it a strong candidate for the venue of decentralized economies and business ecosystems across industries. Nevertheless, Algorand's throughput is still far from…
Reaching agreement in the presence of arbitrary faults is a fundamental problem in distributed computation, which has been shown to be unsolvable if one-third of the processes can fail, unless signed messages are used. In this paper, we propose a solution to a variation of the original BA problem, called Detectable Byzantine Agreement (DBA), that does not need to use signed messages. The proposed algorithm uses what we call $Q$-correlated lists, which are generated by a quantum source device. Once each process has one of these lists, they use them to reach the agreement in a classical manner. …
We give an introduction to federated Byzantine agreement systems (FBAS) with many examples ranging from small "academic" cases to the current Stellar network. We then analyze the main concepts from a mathematical and an algorithmic point of view. Based on work of Lachowski we derive algorithms for quorum enumeration, checking quorum intersection, and computing the intact nodes with respect to a given set of ill-behaved (Byzantine) nodes. We also show that from the viewpoint of the intactness probability of nodes, which we introduce in this paper, a hierarchical setup of nodes is inferior to an…
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.…
In this paper, we study the Byzantine lattice agreement problem in synchronous systems. The lattice agreement problem in crash failure model has been studied both in synchronous and asynchronous systems, which leads to the current best upper bound of $O(\log f)$ rounds in both systems. However, very few algorithmic results are known for the lattice agreement problem in Byzantine failure model. The paper [Nowak et al., DISC, 2019] first gives an algorithm for a variant of the lattice agreement problem on cycle-free lattices that tolerates up to $f < n/(h(X) + 1)$ Byzantine faults, where $n$ is …
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…
We describe an algorithm for Byzantine agreement that is scalable in the sense that each processor sends only $\tilde{O}(\sqrt{n})$ bits, where $n$ is the total number of processors. Our algorithm succeeds with high probability against an \emph{adaptive adversary}, which can take over processors at any time during the protocol, up to the point of taking over arbitrarily close to a 1/3 fraction. We assume synchronous communication but a \emph{rushing} adversary. Moreover, our algorithm works in the presence of flooding: processors controlled by the adversary can send out any number of messages.…
The Byzantine agreement problem requires a set of $n$ processes to agree on a value sent by a transmitter, despite a subset of $b$ processes behaving in an arbitrary, i.e. Byzantine, manner and sending corrupted messages to all processes in the system. It is well known that the problem has a solution in a (an eventually) synchronous message passing distributed system iff the number of processes in the Byzantine subset is less than one third of the total number of processes, i.e. iff $n > 3b+1$. The rest of the processes are expected to be correct: they should never deviate from the algorithm a…
Byzantine agreement algorithms typically assume implicit initial state consistency and synchronization among the correct nodes and then operate in coordinated rounds of information exchange to reach agreement based on the input values. The implicit initial assumptions enable correct nodes to infer about the progression of the algorithm at other nodes from their local state. This paper considers a more severe fault model than permanent Byzantine failures, one in which the system can in addition be subject to severe transient failures that can temporarily throw the system out of its assumption b…
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…
In extending fast digital clock synchronization to the bounded-delay model, the expected constant time Byzantine pulse resynchronization problem is investigated. In this problem, the synchronized state of the system should not only be deterministically maintained but be reached from arbitrary states with expected constant time in the presence of Byzantine faults. An intuitive geometric representation of the problem is introduced, with which the classical approximate agreement, randomized Byzantine agreement, and random walk are integrated with some geometric operations. Efficient realizations …
It has been known since the early 1980s that Byzantine Agreement in the full information, asynchronous model is impossible to solve deterministically against even one crash fault [FLP85], but that it can be solved with probability 1 [Ben83], even against an adversary that controls the scheduling of all messages and corrupts up to $f<n/3$ players [Bra87]. The main downside of [Ben83, Bra87] is that they terminate in $2^{Θ(n)}$ rounds in expectation whenever $f=Θ(n)$. King and Saia [KS16, KS18(arXiv:1812.10169)] developed a polynomial protocol (polynomial rounds, polynomial computation) that is …
In this paper we address Approximate Agreement problem in the Mobile Byzantine faults model. Our contribution is threefold. First, we propose the the first mapping from the existing variants of Mobile Byzantine models to the Mixed-Mode faults model.This mapping further help us to prove the correctness of class MSR (Mean-Subsequence-Reduce) Approximate Agreement algorithms in the Mobile Byzantine fault model, and is of independent interest. Secondly, we prove lower bounds for solving Approximate Agreement under all existing Mobile Byzantine faults models. Interestingly, these lower bounds are d…
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…
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…
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…
In the wake of the decisive impossibility result of Fischer, Lynch, and Paterson for deterministic consensus protocols in the aynchronous model with just one failure, Ben-Or and Bracha demonstrated that the problem could be solved with randomness, even for Byzantine failures. Both protocols are natural and intuitive to verify, and Bracha's achieves optimal resilience. However, the expected running time of these protocols is exponential in general. Recently, Kapron, Kempe, King, Saia, and Sanwalani presented the first efficient Byzantine agreement algorithm in the asynchronous, full information…
We consider the problem of approximate consensus in mobile networks containing Byzantine nodes. We assume that each correct node can communicate only with its neighbors and has no knowledge of the global topology. As all nodes have moving ability, the topology is dynamic. The number of Byzantine nodes is bounded by f and known by all correct nodes. We first introduce an approximate Byzantine consensus protocol which is based on the linear iteration method. As nodes are allowed to collect information during several consecutive rounds, moving gives them the opportunity to gather more values. We …
In the Lattice Agreement (LA) problem, originally proposed by Attiya et al. \cite{Attiya:1995}, a set of processes has to decide on a chain of a lattice. More precisely, each correct process proposes an element $e$ of a certain join-semi lattice $L$ and it has to decide on a value that contains $e$. Moreover, any pair $p_i,p_j$ of correct processes has to decide two values $dec_i$ and $dec_j$ that are comparable (e.g., $dec_i \leq dec_j$ or $dec_j < dec_i$). LA has been studied for its practical applications, as example it can be used to implement a snapshot objects \cite{Attiya:1995} or a rep…
The federated Byzantine agreement system (FBAS) is a consensus model introduced by Mazières in 2016 where the participating nodes conceptually form a network, with links between them being established by each node individually and thus in a decentralized way. An important question is whether these decentralized decisions lead to an overall decentralized network. The level of (de-)centralization in a network can be assessed using centrality measures. In this paper we consider three different approaches for obtaining centrality measures for the nodes in an FBAS. Two of them are based on adapting…
The paper investigates the Lattice Agreement (LA) problem in asynchronous systems. In LA each process proposes an element $e$ from a predetermined lattice, and has to decide on an element $e'$ of the lattice such that $e \leq e'$. Moreover, decisions of different processes have to be comparable (no two processes can decide two elements $e$ and $e'$ such that $(e \not\leq e') \land (e' \not\leq e)$). It has been shown that Generalized LA (i.e., a version of LA proposing and deciding on sequences of values) can be used to build a Replicated State Machine (RSM) with commutative update operations.…
This paper presents a proof of correctness of an iterative approximate Byzantine consensus (IABC) algorithm for directed graphs. The iterative algorithm allows fault- free nodes to reach approximate conensus despite the presence of up to f Byzantine faults. Necessary conditions on the underlying network graph for the existence of a correct IABC algorithm were shown in our recent work [15, 16]. [15] also analyzed a specific IABC algorithm and showed that it performs correctly in any network graph that satisfies the necessary condition, proving that the necessary condition is also sufficient. In…
In this paper we will present the Multidimensional Byzantine Agreement (MBA) Protocol, a leaderless Byzantine agreement protocol defined for complete and synchronous networks that allows a network of nodes to reach consensus on a vector of relevant information regarding a set of observed events. The consensus process is carried out in parallel on each component, and the output is a vector whose components are either values with wide agreement in the network (even if no individual node agrees on every value) or a special value $\bot$ that signals irreconcilable disagreement. The MBA Protocol is…
We address the problem of reaching consensus in the presence of Byzantine faults. In particular, we are interested in investigating the impact of messages relay on the network connectivity for a correct iterative approximate Byzantine consensus algorithm to exist. The network is modeled by a simple directed graph. We assume a node can send messages to another node that is up to $l$ hops away via forwarding by the intermediate nodes on the routes, where $l\in \mathbb{N}$ is a natural number. We characterize the necessary and sufficient topological conditions on the network structure. The tight …
Predictions Can Only Help! Communication Efficient Byzantine Agreement with Predictions
ETH Zürich, Switzerland（苏黎世联邦理工学院）；École Polytechnique Fédérale de Lausanne, Switzerland（洛桑联邦理工学院）；National University of Singapore, Singapore（新加坡国立大学）
Lower bounds and impossibility results in distributed computing are both intellectually challenging and practically important. Hundreds if not thousands of proofs appear in the literature, but surprisingly, the vast majority of them apply to deterministic algorithms only. Probabilistic protocols have been around for at least four decades and are receiving a lot of attention with the emergence of blockchain systems. Nonetheless, we are aware of only a handful of randomized lower bounds. In this paper we provide a formal framework for reasoning about randomized distributed algorithms. We general…
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…
