Byzantine Consensus is Θ(n2): The Dolev-Reischuk Bound is Tight even in Partial Synchrony!

Civit, Pierre; Ayaz, Dzulfikar, Muhammad; Gilbert, Seth; Gramoli, Vincent; Guerraoui, Rachid; Komatovic, Jovan; Vidigueira, Manuel

doi:10.21203/rs.3.rs-2526543/v1

Cited by 6 publications

(17 citation statements)

References 30 publications

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“…The many faces of validity. The literature contains many flavors of validity [4,23,24,28,36,45,59,73,74,78]. One of the most studied properties is Strong Validity [4,23,28,45], stipulating that, if all correct processes propose the same value, only that value can be decided.…”

Section: Introductionmentioning

confidence: 99%

“…The literature contains many flavors of validity [4,23,24,28,36,45,59,73,74,78]. One of the most studied properties is Strong Validity [4,23,28,45], stipulating that, if all correct processes propose the same value, only that value can be decided. Another common property is Weak Validity [23,24,78], affirming that, if all processes are correct and propose the same value, that value must be decided.…”

Section: Introductionmentioning

confidence: 99%

“…One of the most studied properties is Strong Validity [4,23,28,45], stipulating that, if all correct processes propose the same value, only that value can be decided. Another common property is Weak Validity [23,24,78], affirming that, if all processes are correct and propose the same value, that value must be decided. In fact, many other variants of the property have been considered [36,59,73,74].…”

Section: Introductionmentioning

confidence: 99%

“…While validity may appear as an inconspicuous property, its exact definition has a big impact on consensus algorithms. For example, the seminal Dolev-Reischuk bound [30] states that any solution to consensus with Strong Validity incurs a quadratic number of messages; it was recently proven that the bound is tight [23,52,62]. In contrast, while there have been several improvements to the performance of consensus with Weak Validity over the last 40 years [23,52,78], the (tight) lower bound on message complexity remains unknown.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the seminal Dolev-Reischuk bound [30] states that any solution to consensus with Strong Validity incurs a quadratic number of messages; it was recently proven that the bound is tight [23,52,62]. In contrast, while there have been several improvements to the performance of consensus with Weak Validity over the last 40 years [23,52,78], the (tight) lower bound on message complexity remains unknown. (Although the bound is conjectured to be the same as for Strong Validity, this has yet to be formally proven.)…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On the Validity of Consensus

Civit¹,

Gilbert²,

Guerraoui³

et al. 2023

Preprint

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The Byzantine consensus problem involves 𝑛 processes, out of which 𝑡 < 𝑛 could be faulty and behave arbitrarily. Three properties characterize consensus: (1) termination, requiring correct (non-faulty) processes to eventually reach a decision, (2) agreement, preventing them from deciding different values, and (3) validity, precluding "unreasonable" decisions. But, what is a reasonable decision? Strong validity, a classical property, stipulates that, if all correct processes propose the same value, only that value can be decided. Weak validity, another established property, stipulates that, if all processes are correct and they propose the same value, that value must be decided. The space of possible validity properties is vast. However, their impact on consensus remains unclear.This paper addresses the question of which validity properties allow Byzantine consensus to be solvable with partial synchrony, and at what cost. First, we determine necessary and sufficient conditions for a validity property to make the consensus problem solvable; we say that such validity properties are solvable. Notably, we prove that, if 𝑛 ≤ 3𝑡, all solvable validity properties are trivial (there exists an always-admissible decision). Furthermore, we show that, with any non-trivial (and solvable) validity property, consensus requires Ω(𝑡 2 ) messages. This extends the seminal Dolev-Reischuk bound, originally proven for strong validity, to all non-trivial validity properties. Lastly, we give a general Byzantine consensus algorithm, we call Universal, for any solvable (and non-trivial) validity property. Importantly, Universal incurs 𝑂 (𝑛 2 ) message complexity. Thus, together with our lower bound, Universal implies a fundamental result in partial synchrony: with 𝑡 ∈ Ω(𝑛), the message complexity of all (non-trivial) consensus variants is Θ(𝑛 2 ).

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Validity of Consensus

Civit¹,

Gilbert²,

Guerraoui³

et al. 2023

Preprint

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Byzantine consensus is $$\Theta (n^2)$$: the Dolev-Reischuk bound is tight even in partial synchrony!

Civit,

Dzulfikar,

Gilbert

et al. 2023

Distrib. Comput.

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The Dolev-Reischuk bound says that any deterministic Byzantine consensus protocol has (at least) quadratic (in the number of processes) communication complexity in the worst case: given a system with n processes and at most $$f < n / 3$$ f < n / 3 failures, any solution to Byzantine consensus exchanges $$\Omega \big (n^2\big )$$ Ω ( n 2 ) words, where a word contains a constant number of values and signatures. While it has been shown that the bound is tight in synchronous environments, it is still unknown whether a consensus protocol with quadratic communication complexity can be obtained in partial synchrony where the network alternates between (1) asynchronous periods, with unbounded message delays, and (2) synchronous periods, with $$\delta $$ δ -bounded message delays. Until now, the most efficient known solutions for Byzantine consensus in partially synchronous settings had cubic communication complexity (e.g., HotStuff, binary DBFT). This paper closes the existing gap by introducing SQuad, a partially synchronous Byzantine consensus protocol with $$O\big (n^2\big )$$ O ( n 2 ) worst-case communication complexity. In addition, SQuad is optimally-resilient (tolerating up to $$f < n / 3$$ f < n / 3 failures) and achieves $$O(f \cdot \delta )$$ O ( f · δ ) worst-case latency complexity. The key technical contribution underlying SQuad lies in the way we solve view synchronization, the problem of bringing all correct processes to the same view with a correct leader for sufficiently long. Concretely, we present RareSync, a view synchronization protocol with $$O\big (n^2\big )$$ O ( n 2 ) communication complexity and $$O(f \cdot \delta )$$ O ( f · δ ) latency complexity, which we utilize in order to obtain SQuad.

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Expected linear round synchronization: the missing link for linear Byzantine SMR

Naor,

Keidar

2024

Distrib. Comput.

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State Machine Replication (SMR) solutions often divide time into rounds, with a designated leader driving decisions in each round. Progress is guaranteed once all correct processes synchronize to the same round, and the leader of that round is correct. Recently suggested Byzantine SMR solutions such as HotStuff, Tendermint, and LibraBFT achieve progress with a linear message complexity and a constant time complexity once such round synchronization occurs. But round synchronization itself incurs an additional cost. By Dolev and Reischuk's lower bound, any deterministic solution must have Ω(n 2 ) communication complexity. Yet the question of randomized round synchronization with an expected linear message complexity remained open.We present an algorithm that, for the first time, achieves round synchronization with expected linear message complexity and expected constant latency. Existing protocols can use our round synchronization algorithm to solve Byzantine SMR with the same asymptotic performance.

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Byzantine Consensus is Θ(n2): The Dolev-Reischuk Bound is Tight even in Partial Synchrony!

Cited by 6 publications

References 30 publications

On the Validity of Consensus

On the Validity of Consensus

Byzantine consensus is $$\Theta (n^2)$$: the Dolev-Reischuk bound is tight even in partial synchrony!

Expected linear round synchronization: the missing link for linear Byzantine SMR

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