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Broadcast Amplification
"... Abstract. A dbroadcast primitive is a communication primitive that allows a sender to send a value from a domain of size d to a set of parties. A broadcast protocol emulates the dbroadcast primitive using only pointtopoint channels, even if some of the parties cheat, in the sense that all correc ..."
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Abstract. A dbroadcast primitive is a communication primitive that allows a sender to send a value from a domain of size d to a set of parties. A broadcast protocol emulates the dbroadcast primitive using only pointtopoint channels, even if some of the parties cheat, in the sense that all correct recipients agree on the same value v (consistency), and if the sender is correct, then v is the value sent by the sender (validity). A celebrated result by Pease, Shostak and Lamport states that such a broadcast protocol exists if and only if t < n/3, where n denotes the total number of parties and t denotes the upper bound on the number of cheaters. This paper is concerned with broadcast protocols for any number of cheaters (t < n), which can be possible only if, in addition to pointtopoint channels, another primitive is available. Broadcast amplification is the problem of achieving dbroadcast when d ′broadcast can be used once, for d ′ < d. Let φn(d) denote the minimal such d ′ for domain size d. We show that for n = 3 parties, broadcast for any domain size is possible if only a single 3broadcast is available, and broadcast of a single bit (d ′ = 2) is not sufficient, i.e., φ3(d) = 3 for any d ≥ 3. In contrast, for n> 3 no broadcast amplification is possible, i.e., φn(d) = d for any d. However, if other parties than the sender can also broadcast some short messages, then broadcast amplification is possible for any n. Let φ ∗ n(d) denote the minimal d ′ such that dbroadcast can be constructed from primitives d ′ 1broadcast,..., d ′ kbroadcast, where d ′ = ∏ i d ′ i (i.e., log d ′ ∑ i log d ′ i). Note that φ ∗ n(d) ≤ φn(d). We show that broadcasting 8n log n bits in total suffices, independently of d, and that at least n−2 parties, including the sender, must broadcast at least one bit. Hence min(log d, n − 2) ≤ log φ ∗ n(d) ≤ 8n log n.
EarlyDeciding Consensus is Expensive
"... In consensus, the n nodes of a distributed system seek to take a consistent decision on some output, despite up to t of them crashing or even failing maliciously, i.e., behaving “Byzantine”. It is known that it is impossible to guarantee that synchronous, deterministic algorithms consistently decide ..."
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In consensus, the n nodes of a distributed system seek to take a consistent decision on some output, despite up to t of them crashing or even failing maliciously, i.e., behaving “Byzantine”. It is known that it is impossible to guarantee that synchronous, deterministic algorithms consistently decide on an output in fewer than f + 1 rounds in executions in which the actual number of faults is f ≤ t. This even holds if faults are crashonly, and in this case the bound can be matched precisely. However, the question of whether this can be done efficiently, i.e., with little communication, so far has not been addressed. In this work, we show that algorithms tolerating Byzantine faults and deciding within f + 2 rounds must send Ω(nt + t 2 f) messages; as a byproduct, our analysis shows that decision within f +1 rounds is impossible in this setting (unless f = t). Moreover, we prove that any crashresilient algorithm deciding in f + 1 rounds has worstcase message complexity Ω(n 2 f). Interestingly, this changes drastically if we restrict the fault model further. If crashes are orderly, i.e., in each round, each node picks an order in which its messages are sent, and crashing nodes successfully transmit a prefix of their sequence, deciding in f + 1 rounds can be guaranteed with O(nt) messages.
MultiValued Byzantine Broadcast: the t < n Case
"... All known protocols implementing broadcast from synchronous pointtopoint channels tolerating any t < n Byzantine corruptions have communication complexity at least Ω(ℓn 2). We give cryptographically secure and informationtheoretically secure protocols for t < n that communicate O(ℓn) bits i ..."
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All known protocols implementing broadcast from synchronous pointtopoint channels tolerating any t < n Byzantine corruptions have communication complexity at least Ω(ℓn 2). We give cryptographically secure and informationtheoretically secure protocols for t < n that communicate O(ℓn) bits in order to broadcast sufficiently long ℓ bit messages. This matches the optimal communication complexity bound for any protocol allowing to broadcast ℓ bit messages. While broadcast protocols with the optimal communication complexity exist in cases where t < n/3 (by Liang and Vaidya in PODC ’11) or t < n/2 (by Fitzi and Hirt in PODC ’06), this paper is the first to present such protocols for t < n.
Asynchronous Byzantine Systems: From Multivalued to Binary Consensus with t < n/3, O(n2) Messages, O(1) Time, and no Signature
, 2015
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
MultiValued Byzantine Broadcast: the t < n Case Anonymous submission
"... Abstract. Byzantine broadcast is a distributed primitive that allows a specific party to consistently distribute a message among n parties in the presence of potential misbehavior of up to t of the parties. All known protocols implementing broadcast of an ℓbit message from pointtopoint channels t ..."
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Abstract. Byzantine broadcast is a distributed primitive that allows a specific party to consistently distribute a message among n parties in the presence of potential misbehavior of up to t of the parties. All known protocols implementing broadcast of an ℓbit message from pointtopoint channels tolerating any t < n Byzantine corruptions have communication complexity at least Ω(ℓn 2). In this paper we give cryptographically secure and informationtheoretically secure protocols for t < n that communicate O(ℓn) bits when ℓ is sufficiently large. This matches the optimal communication complexity bound for any protocol allowing to broadcast ℓbit messages. While broadcast protocols with the optimal communication complexity exist for t < n/2, this paper is the first to present such protocols for t < n.