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Partial Synchrony for Free? New Upper Bounds for Byzantine Agreement

Pierre Civit, Muhammad Ayaz Dzulfikar, Seth Gilbert, Rachid Guerraoui, Jovan Komatovic, Manuel Vidigueira, Igor Zablotchi

TL;DR

This paper introduces Oper, the first generic transformation of deterministic Byzantine agreement algorithms from synchrony to partial synchrony, and presents the first partially synchronous Byzantine agreement algorithm that achieves optimal O(n^2) bit complexity, requires no cryptography, and is optimally resilient, thus showing that the Dolev-Reischuk bound is tight even in partial synchrony.

Abstract

Byzantine agreement allows n processes to decide on a common value, in spite of arbitrary failures. The seminal Dolev-Reischuk bound states that any deterministic solution to Byzantine agreement exchanges Omega(n^2) bits. In synchronous networks, solutions with optimal O(n^2) bit complexity, optimal fault tolerance, and no cryptography have been established for over three decades. However, these solutions lack robustness under adverse network conditions. Therefore, research has increasingly focused on Byzantine agreement for partially synchronous networks. Numerous solutions have been proposed for the partially synchronous setting. However, these solutions are notoriously hard to prove correct, and the most efficient cryptography-free algorithms still require O(n^3) exchanged bits in the worst case. In this paper, we introduce Oper, the first generic transformation of deterministic Byzantine agreement algorithms from synchrony to partial synchrony. Oper requires no cryptography, is optimally resilient (n >= 3t+1, where t is the maximum number of failures), and preserves the worst-case per-process bit complexity of the transformed synchronous algorithm. Leveraging Oper, we present the first partially synchronous Byzantine agreement algorithm that (1) achieves optimal O(n^2) bit complexity, (2) requires no cryptography, and (3) is optimally resilient (n >= 3t+1), thus showing that the Dolev-Reischuk bound is tight even in partial synchrony. Moreover, we adapt Oper for long values and obtain several new partially synchronous algorithms with improved complexity and weaker (or completely absent) cryptographic assumptions.

Partial Synchrony for Free? New Upper Bounds for Byzantine Agreement

TL;DR

This paper introduces Oper, the first generic transformation of deterministic Byzantine agreement algorithms from synchrony to partial synchrony, and presents the first partially synchronous Byzantine agreement algorithm that achieves optimal O(n^2) bit complexity, requires no cryptography, and is optimally resilient, thus showing that the Dolev-Reischuk bound is tight even in partial synchrony.

Abstract

Byzantine agreement allows n processes to decide on a common value, in spite of arbitrary failures. The seminal Dolev-Reischuk bound states that any deterministic solution to Byzantine agreement exchanges Omega(n^2) bits. In synchronous networks, solutions with optimal O(n^2) bit complexity, optimal fault tolerance, and no cryptography have been established for over three decades. However, these solutions lack robustness under adverse network conditions. Therefore, research has increasingly focused on Byzantine agreement for partially synchronous networks. Numerous solutions have been proposed for the partially synchronous setting. However, these solutions are notoriously hard to prove correct, and the most efficient cryptography-free algorithms still require O(n^3) exchanged bits in the worst case. In this paper, we introduce Oper, the first generic transformation of deterministic Byzantine agreement algorithms from synchrony to partial synchrony. Oper requires no cryptography, is optimally resilient (n >= 3t+1, where t is the maximum number of failures), and preserves the worst-case per-process bit complexity of the transformed synchronous algorithm. Leveraging Oper, we present the first partially synchronous Byzantine agreement algorithm that (1) achieves optimal O(n^2) bit complexity, (2) requires no cryptography, and (3) is optimally resilient (n >= 3t+1), thus showing that the Dolev-Reischuk bound is tight even in partial synchrony. Moreover, we adapt Oper for long values and obtain several new partially synchronous algorithms with improved complexity and weaker (or completely absent) cryptographic assumptions.
Paper Structure (51 sections, 102 theorems, 10 equations, 5 figures, 10 tables, 15 algorithms)

This paper contains 51 sections, 102 theorems, 10 equations, 5 figures, 10 tables, 15 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Oper: Overview
  4. Key Idea
  5. Implementation of the Key Idea
  6. Interpretation of Oper's Views via Spider Graphs
  7. Structure of Oper's Views.
  8. Preliminaries
  9. Crux: The View Logic of Oper
  10. Crux's Specification
  11. Crux's Building Blocks
  12. Graded Consensus (Module \ref{['mod:graded-consensus']})
  13. Validation Broadcast (Module \ref{['mod:validation-broadcast']})
  14. Crux's Pseudocode
  15. From Crux to Oper
  16. ...and 36 more sections

Key Result

Theorem 1.1

Given any $t$-resilient $(t < n / 3)$ deterministic synchronous Byzantine agreement algorithm $\mathcal{A}^S$ with worst-case per-process bit complexity $\mathcal{B}$ and worst-case latency $\mathcal{L}$, $\textsc{Oper}\xspace(\mathcal{A}^S)$ is a $t$-resilient deterministic partially synchronous By

Figures (5)

  • Figure 1: Interpration of Oper's views using spider graphs for $|\mathsf{Value}| = 4$. In \ref{['fig:spider-graph']}, all possible positions are represented with circles. \ref{['fig:divergent_config', 'fig:convergent_config']} illustrate examples of divergent and convergent configurations, respectively; each process is represented with a circle.
  • Figure 2: Structure of a view in Oper.
  • Figure 3: Illustration of the preservation of (initial) convergence in a view.
  • Figure 4: Illustration of a view achieving convergence after GST.
  • Figure 5: Overview of Crux.

Theorems & Definitions (106)

  • Theorem 1.1: Main
  • Lemma A .1
  • Lemma A .2
  • Theorem 1.1: Strong validity
  • Theorem 1.2: External validity
  • Theorem 1.3: Agreement
  • Theorem 1.4: Integrity
  • Theorem 1.5: Termination
  • Theorem 1.6: Totality
  • Theorem 1.7: Completion time
  • ...and 96 more