Consider a fully-connected synchronous distributed system consisting of @math nodes, where up to @math nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous @math -counting problem, all nodes need to eventually agree on a counter that is increased by one modulo @math in each round for given @math . In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external "go" signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a "fire" signal. Moreover, no node should generate a "fire" signal without some correct node having previously received a "go" signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience @math , asymptotically optimal stabilisation and response time @math , and message size @math . As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions, and it is straightforward to adapt our framework for other types of permanent faults.
In a seminal work, Dolev and Welch @cite_18 showed that the task can be solved in a self-stabilising manner in the presence of (the optimal number of) @math Byzantine faults using randomisation; see also [Ch. 6] dolev00self-stabilization . While this algorithm can be implemented using only constant-size messages, the expected stabilisation time is exponential. Later, Ben- @cite_13 showed that it is possible to obtain optimally-resilient solutions that stabilise in expected constant time. However, their algorithm relies on shared coins, which are costly to implement and assume private communication channels.
@cite_18 @cite_13
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In this paper, we show how to build synchronized clocks of arbitrary size atop of existing small-sized clocks, despite arbitrary faults. Our solution is both self-stabilizing and Byzantine fault-tolerant, and needs merely single-bit channels. It involves a reduction to Byzantine fault-tolerant consensus, which allows different consensus algorithms to be plugged in for matching the actual clock sizes and resilience requirements best. We demonstrate the practicability of our approach by means of an FPGA implementation and its experimental evaluation. To also address the cases where deterministic algorithms hit fundamental limits, we provide a novel randomized self-stabilizing Byzantine consensus algorithm that works very well also in these settings, along with its correctness proof and stabilization time analysis. Consider a complete communication network of @math nodes, where the nodes receive a common clock pulse. We study the synchronous @math -counting problem: given any starting state and up to @math faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo @math in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilisation time in @math (asymptotically optimal), (4) use a small number of states, and consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomisation, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilisation. Consider a distributed network in which up to a third of the nodes may be Byzantine, and in which the non-faulty nodes may be subject to transient faults that alter their memory in an arbitrary fashion. Within the context of this model, we are interested in the digital clock synchronization problem; which consists of agreeing on bounded integer counters, and increasing these counters regularly. It has been postulated in the past that synchronization cannot be solved in a Byzantine tolerant and self-stabilizing manner. The first solution to this problem had an expected exponential convergence time. Later, a deterministic solution was published with linear convergence time, which is optimal for deterministic solutions. In the current paper we achieve an expected constant convergence time. We thus obtain the optimal probabilistic solution, both in terms of convergence time and in terms of resilience to Byzantine adversaries. Awareness of the need for robustness in distributed systems increases as distributed systems become integral parts of day-to-day systems. Self-stabilizing while tolerating ongoing Byzantine faults are wishful properties of a distributed system. Many distributed tasks (e.g. clock synchronization) possess efficient non-stabilizing solutions tolerating Byzantine faults or conversely non-Byzantine but self-stabilizing solutions. In contrast, designing algorithms that self-stabilize while at the same time tolerating an eventual fraction of permanent Byzantine failures present a special challenge due to the “ambition” of malicious nodes to hamper stabilization if the systems tries to recover from a corrupted state. This difficulty might be indicated by the remarkably few algorithms that are resilient to both fault models. We present the first scheme that takes a Byzantine distributed algorithm and produces its self-stabilizing Byzantine counterpart, while having a relatively low overhead of O(f′) communication rounds, where f′ is the number of actual faults. Our protocol is based on a tight Byzantine self-stabilizing pulse synchronization procedure. The synchronized pulses are used as events for initializing Byzantine agreement on every node’s local state. The set of local states is used for global predicate detection. Should the global state represent an illegal system state then the target algorithm is reset.