We consider the problem of making distributed computations robust to noise, in particular to worst-case (adversarial) corruptions of messages. We give a general distributed interactive coding scheme which simulates any asynchronous distributed protocol while tolerating an optimal corruption of a @math fraction of all messages while incurring a moderate blowup of @math in the communication complexity. Our result is the first fully distributed interactive coding scheme in which the topology of the communication network is not known in advance. Prior work required either a coordinating node to be connected to all other nodes in the network or assumed a synchronous network in which all nodes already know the complete topology of the network.
Pelc @cite_5 shows that if the number of Byzantine-corrupted links is bounded by @math , reliable communication is achievable only over graphs whose connectivity is more than @math . The same work also considers the case where each link is faulty with some probability. In a more recent work, Feinerman, Haeupler and Korman @cite_36 also address complete synchronous networks, and study the specific problems of broadcast and majority consensus under random errors.
@cite_36 @cite_5
2071898078 2950535145 1980514553 75522826
We construct and analyze a fast broadcasting algorithm working in the presence of Byzantine component faults. Such faults are particularly difficult to deal with, as faulty components may behave arbitrarily (even maliciously) as transmitters, by either blocking, rerouting, or altering transmitted messages in a way most detrimental to the broadcasting process. We assume that links and nodes of a communication network are subject to Byzantine failures, and that faults are distributed randomly and independently, with link failure probability p and node failure probability q, these parameters being constant and satisfying the inequality (1 - p)2(1 - q) > 1 2. A broadcasting algorithm, working in an n-node network, is called almost safe if the probability of its correctness is at least 1 - 1 n, for sufficiently large n. Thus the robustness of the algorithm grows with the size of the network. Our main result is the design and analysis of an almost safe broadcasting algorithm working in time O(log2 n) and using O(n log n) messages in n-node networks. Under a stronger assumption on failure probability parameters, namely (1 - p)2(1 - q)2 > 1 2, our algorithm can be modified to work in time O(log2 n log log n), also using O(n log n) messages. The novelty of our algorithm is that it can cope with the most difficult type of faults, potentially affecting all components of the network (both its links and nodes), and that it is simultaneously robust and efficient. 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 Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern. In this paper we study the problem of achieving Byzantine agreement among a set of processors, where the processors are computationally sound but their interfaces with the communication channels may be faulty. We consider three types of fault, namely message corruption, message loss, and spurious message generation. We present the following results for this model: (i) If all three types of faults are present then the problem is equivalent to the classical Byzantine generals problem. (ii) In the cases where only message corruption can occur, agreement becomes trivial and can be achieved in one round. (iii) If spurious message generation is ruled out, that is, when interfaces may fault only when sensitized, agreement is possible irrespective of the ratio of the number of processors having faulty interfaces with the total number of processors. 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.