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.... We will state a consequence of it here in the context of Hamming distance, as we will not need the additional generality provided by the full statement. A fuller statement and proof can be found in =-=[2]-=-, for example. 9 We first develop some convenient notation. We let Ω = ∏n i=1Ωi, where each Ωi is a probability space and Ω is endowed with the product measure. We employ the usual notion of Hamming d...

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... failures, which cause a processor to quit without warning, as well as resetting failures, which we will define and motivate below. In this setting, the elegant result of Fischer, Lynch, and Paterson =-=[14]-=- shows that it is already impossible to design a deterministic protocol for agreement that always terminates, even if the adversary is limited to causing at most one crash failure. A common approach f...

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...unds and other relevant work. 1.1 Related Work The problem of reaching agreement despite faults was introduced by Pease, Shostak, and Lamport in [26], who also proposed the Byzantine failure model in =-=[23]-=-. Since its introduction, the problem of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which ...

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...e in practice is to use an algorithm that terminates as long as worst-case scheduling does not occur indefinitely. This is a property achieved by the well-known Paxos algorithm constructed by Lamport =-=[22]-=-. Randomized algorithms provide a potential alternative. Quickly 1 following the impossibility result, Ben-Or [7] and Bracha [10] presented randomized algorithms terminating with probability one, even...

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... following subsection, we briefly survey prior lower bounds and other relevant work. 1.1 Related Work The problem of reaching agreement despite faults was introduced by Pease, Shostak, and Lamport in =-=[26]-=-, who also proposed the Byzantine failure model in [23]. Since its introduction, the problem of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered c...

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...tive Byzantine adversary and the strongly adaptive adversary. Our Techniques To prove the exponential lower bound on running time, we rely crucially on a general probabilistic inequality of Talagrand =-=[28]-=-, which roughly states that any product distribution Ω1×Ω2× . . .×Ωn cannot put too much weight simultaneously on two sets A and B in n-dimensional space that are “far apart.” For our purposes, “far a...

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... This is a property achieved by the well-known Paxos algorithm constructed by Lamport [22]. Randomized algorithms provide a potential alternative. Quickly 1 following the impossibility result, Ben-Or =-=[7]-=- and Bracha [10] presented randomized algorithms terminating with probability one, even against such strong adversaries. These algorithms were intuitively structured, and Bracha’s algorithm tolerated ...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...

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...s at least 1/ck for some constant c. Their technique involves constructing a chain of indistinguishable executions and bounding the termination probability in terms of the length of the chain. Aspnes =-=[3]-=- proves a lower bound of Ω(t/ log2 t) on the expected number of local coin flips for asynchronous algorithms against adaptive adversaries that holds in either the shared memory or message passing mode...

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...[27, 29, 9, 12, 11, 25], for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in =-=[20, 21, 8, 15]-=-. Recent work has focused on reducing the communication overhead of synchronous protocols [17, 18]. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in...

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...erty achieved by the well-known Paxos algorithm constructed by Lamport [22]. Randomized algorithms provide a potential alternative. Quickly 1 following the impossibility result, Ben-Or [7] and Bracha =-=[10]-=- presented randomized algorithms terminating with probability one, even against such strong adversaries. These algorithms were intuitively structured, and Bracha’s algorithm tolerated an optimal numbe...

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...[27, 29, 9, 12, 11, 25], for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in =-=[20, 21, 8, 15]-=-. Recent work has focused on reducing the communication overhead of synchronous protocols [17, 18]. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in...

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...[27, 29, 9, 12, 11, 25], for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in =-=[20, 21, 8, 15]-=-. Recent work has focused on reducing the communication overhead of synchronous protocols [17, 18]. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in...

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...ized setting. In the shared memory model, there are polynomial time randomized algorithms tolerating crash failures, and tight bounds on their total step complexity are proven by Attiya and Censor in =-=[5]-=-. This work also uses an analysis of product probability spaces, similarly to the proof in [6]. 2 Models and Definitions We let n denote the total number of processors, and consider each processor to ...

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...rithms, but this class inherently limits the amount of randomness used in choosing an individual message. Polynomial lower bounds for randomized algorithms include the result of Bar-Joseph and Ben-Or =-=[6]-=-, which proves a lower bound of t/ √ n log n on the number of expected rounds for a randomized synchronous protocol against an adversary who can adaptively choose to fail t processors. Interestingly, ...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...

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...[27, 29, 9, 12, 11, 25], for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in =-=[20, 21, 8, 15]-=-. Recent work has focused on reducing the communication overhead of synchronous protocols [17, 18]. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in...

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...ly adaptive adversary. This provides a rather complete understanding of what is achievable in the presence of adaptive resetting faults. In contrast, the relatively recent algorithm of Kapron et. al. =-=[16]-=- runs very quickly (polylogarithmic time in n) and tolerates t < ( 1 3 − ǫ) non-adaptive Byzantine failures, but incurs a non-zero probability of non-termination or termination with invalid outputs. I...

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...lipping games, similar to our core use of Talagrand’s inequality. However, they employ different techniques to build their analysis of multiple rounds. Another lower bound is due to Attiya and Censor =-=[4]-=-, who show that for any integer k, the probability that a randomized Byzantine agreement algorithm does not terminate in k(n − t) steps is at least 1/ck for some constant c. Their technique involves c...

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...ndomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent work has focused on reducing the communication overhead of synchronous protocols =-=[17, 18]-=-. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in the asynchronous setting mentioned above, there is a sharp lower bound of t rounds for determinis...

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...ndomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent work has focused on reducing the communication overhead of synchronous protocols =-=[17, 18]-=-. Several lower bounds are also known. In addition to the impossibility of deterministic algorithms in the asynchronous setting mentioned above, there is a sharp lower bound of t rounds for determinis...

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...ly structured, and Bracha’s algorithm tolerated an optimal number of failures, namely allowing for t processors to behave in an arbitrary malicious fashion, for any t < n 3 . Also, Aguilera and Toueg =-=[1]-=- have provided a new correctness proof for Ben-Or’s randomized consensus algorithm when there are < n 2 crash failures. However, for some settings of the initial input bits, the algorithms of [7, 10] ...

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...n on the support size as in [24]. We also avoid any requirement of symmetry in how received messages are treated, and our class is more intuitively defined. Concurrently with this work, King and Saia =-=[19]-=- have discovered a Las Vegas polynomialtime algorithm tolerating adaptive Byzantine faults that falls outside the classes of algorithms considered here and in [24]. This implies a separation between w...

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...a nonzero chance that the final committee is composed entirely of faulty processors. With the goal of beginning a systematic study of what can be achieved without incurring these disadvantages, Lewko =-=[24]-=- previously proved that a class of algorithms generalizing Ben-Or and Bracha’s algorithms in [7, 10] cannot achieve subexponential running time against an adversary causing t = Ω(n) non2 adaptive Byza...

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...addition to the impossibility of deterministic algorithms in the asynchronous setting mentioned above, there is a sharp lower bound of t rounds for deterministic algorithms in the synchronous setting =-=[13]-=-. This lower bound is proven by assembling a chain of executions where any two adjacent executions are indistinguishable to some non-faulty processor and the two ends of the chain represent different ...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...

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... of fault-tolerant agreement has been widely studied in a variety of models. Several works have considered computationally bounded adversaries, a setting in which cryptographic tools can be employed (=-=[27, 29, 9, 12, 11, 25]-=-, for example). In the synchronous communication setting, polylogarithmic round randomized protocols for Byzantine agreement against 3 non-adaptive adversaries were obtained in [20, 21, 8, 15]. Recent...