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 sucient 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 suciency by presenting a Byzantine consensus algorithm for directed graphs that satisfy the necessary condition.
Lamport, Shostak, and Pease addressed the Byzantine agreement problem in @cite_13 . Subsequent work @cite_0 @cite_12 characterized the necessary and sufficient conditions under which the problem is solvable in undirected graphs. However, as noted above, these conditions are not adequate to fully characterize the directed graphs in which Byzantine consensus is feasible. In this work, we identify tight necessary and sufficient conditions for Byzantine consensus in directed graphs. The necessity proof presented in this paper is based on the state-machine approach, which was originally developed for conditions in undirected graphs @cite_0 @cite_3 @cite_15 ; however, due to the nature of directed links, our necessity proof is a non-trivial extension. The technique is also similar to the withholding mechanism, which was developed by Schmid, Weiss, and Keidar @cite_8 to prove impossibility results and lower bounds for the number of nodes for synchronous consensus under transient link failures in fully-connected graphs; however, we do not assume the transient fault model as in @cite_8 , and thus, our argument is more straightforward.
@cite_8 @cite_3 @cite_0 @cite_15 @cite_13 @cite_12
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Three decades ago, introduced the problem of Byzantine Agreement [PSL80] where nodes need to maintain a consistent view of the world in spite of the challenge posed by Byzantine faults. Subsequently, it is well known that Byzantine agreement over a completely connected synchronous network of n nodes tolerating up to t faults is (efficiently) possible if and only if t < n 3. further empowered the nodes with the ability to authenticate themselves and their messages and proved that agreement in this new model (popularly known as authenticated Byzantine agreement (ABA)) is possible if and only if t < n. (which is a huge improvement over the bound of t < n 3 in the absence of authentication for the same functionality). To understand the utility, potential and limitations of using authentication in distributed protocols for agreement, [GGBS10] studied ABA in new light. They generalize the existing models and thus, attempt to give a unified theory of agreements over the authenticated and non-authenticated domains. In this paper we extend their results to synchronous (undirected) networks and give a complete characterization of agreement protocols. As a corollary, we show that agreement can be strictly easier than all-pair point-to-point communication. It is well known that in a synchronous network over n nodes of which up to any t are corrupted by a Byzantine adversary, BA is possible only if all pair point-to-point reliable communication is possible [Dol82, DDWY93]. Thus, a folklore in the area is that maintaining global consistency (agreement) is at least as hard as the problem of all pair point-to-point communication. Equivalently, it is widely believed that protocols for BA over incomplete networks exist only if it is possible to simulate an overlay-ed complete network. Surprisingly, we show that the folklore is not always true. Thus, it seems that agreement protocols may be more fundamental to distributed computing than reliable communication. This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic execution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous movements and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduction scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems. This work addresses Byzantine vector consensus, wherein the input at each process is a d-dimensional vector of reals, and each process is required to decide on a decision vector that is in the convex hull of the input vectors at the fault-free processes [9,12]. The input vector at each process may also be viewed as a point in the d-dimensional Euclidean space R d , where di¾?>i¾?0 is a finite integer. Recent work [9,12] has addressed Byzantine vector consensus, and presented algorithms with optimal fault tolerance in complete graphs. This paper considers Byzantine vector consensus in incomplete graphs using a restricted class of iterative algorithms that maintain only a small amount of memory across iterations. For such algorithms, we prove a necessary condition, and a sufficient condition, for the graphs to be able to solve the vector consensus problem iteratively. We present an iterative Byzantine vector consensus algorithm, and prove it correct under the sufficient condition. The necessary condition presented in this paper for vector consensus does not match with the sufficient condition for di¾?>i¾?1; thus, a weaker condition may potentially suffice for Byzantine vector consensus. Byzantine Agreement has become increasingly important in establishing distributed properties when errors may exist in the systems. Recent polynomial algorithms for reaching Byzantine Agreement provide us with feasible solutions for obtaining coordination and synchronization in distributed systems. In this paper the amount of information exchange necessary to ensure Byzantine Agreement is studied. This is measured by the total number of messages the participating processors have to send in the worst case. In algorithms that use a signature scheme, the number of signatures appended to messages are also counted. First it is shown that O( nt ) is a lower bound for the number of signatures for any algorithm using authentication, where n denotes the number of processors and t the upper bound on the number of faults the algorithm is supposed to handle. For algorithms that reach Byzantine Agreement without using authentication this is even a lower bound for the total number of messages. If n is large compared to t , these bounds match the upper bounds from previously known algorithms. For the number of messages in the authenticated case we prove the lower bound O( n + t 2 ). Finally algorithms that achieve this bound are presented.