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Fully-Distributed Byzantine Agreement in Sparse Networks

John Augustine, Fabien Dufoulon, Gopal Pandurangan

TL;DR

This work presents fully-distributed Byzantine agreement protocols for sparse, bounded degree networks that tolerate significantly more Byzantine nodes -- up to $O(n/ polylog(n)$ of them.

Abstract

Byzantine agreement is a fundamental problem in fault-tolerant distributed networks that has been studied intensively for the last four decades. Most of these works designed protocols for complete networks. A key goal in Byzantine protocols is to tolerate as many Byzantine nodes as possible. The work of Dwork, Peleg, Pippenger, and Upfal [STOC 1986, SICOMP 1988] was the first to address the Byzantine agreement problem in sparse, bounded degree networks and presented a protocol that achieved almost-everywhere agreement among honest nodes. In such networks, all known Byzantine agreement protocols (e.g., Dwork, Peleg, Pippenger, and Upfal, STOC 1986; Upfal, PODC 1992; King, Saia, Sanwalani, and Vee, FOCS 2006) that tolerated a large number of Byzantine nodes had a major drawback that they were not fully-distributed -- in those protocols, nodes are required to have initial knowledge of the entire network topology. This drawback makes such protocols inapplicable to real-world communication networks such as peer-to-peer (P2P) networks, which are typically sparse and bounded degree and where nodes initially have only local knowledge of themselves and their neighbors. Indeed, a fundamental open question raised by the above works is whether one can design Byzantine protocols that tolerate a large number of Byzantine nodes in sparse networks that work with only local knowledge, i.e., fully-distributed protocols. The work of Augustine, Pandurangan, and Robinson [PODC 2013] presented the first fully-distributed Byzantine agreement protocol that works in sparse networks, but it tolerated only up to $O(\sqrt{n}/ polylog(n))$ Byzantine nodes (where $n$ is the total network size). We answer the earlier open question by presenting fully-distributed Byzantine agreement protocols for sparse, bounded degree networks that tolerate significantly more Byzantine nodes -- up to $O(n/ polylog(n))$ of them.

Fully-Distributed Byzantine Agreement in Sparse Networks

TL;DR

This work presents fully-distributed Byzantine agreement protocols for sparse, bounded degree networks that tolerate significantly more Byzantine nodes -- up to of them.

Abstract

Byzantine agreement is a fundamental problem in fault-tolerant distributed networks that has been studied intensively for the last four decades. Most of these works designed protocols for complete networks. A key goal in Byzantine protocols is to tolerate as many Byzantine nodes as possible. The work of Dwork, Peleg, Pippenger, and Upfal [STOC 1986, SICOMP 1988] was the first to address the Byzantine agreement problem in sparse, bounded degree networks and presented a protocol that achieved almost-everywhere agreement among honest nodes. In such networks, all known Byzantine agreement protocols (e.g., Dwork, Peleg, Pippenger, and Upfal, STOC 1986; Upfal, PODC 1992; King, Saia, Sanwalani, and Vee, FOCS 2006) that tolerated a large number of Byzantine nodes had a major drawback that they were not fully-distributed -- in those protocols, nodes are required to have initial knowledge of the entire network topology. This drawback makes such protocols inapplicable to real-world communication networks such as peer-to-peer (P2P) networks, which are typically sparse and bounded degree and where nodes initially have only local knowledge of themselves and their neighbors. Indeed, a fundamental open question raised by the above works is whether one can design Byzantine protocols that tolerate a large number of Byzantine nodes in sparse networks that work with only local knowledge, i.e., fully-distributed protocols. The work of Augustine, Pandurangan, and Robinson [PODC 2013] presented the first fully-distributed Byzantine agreement protocol that works in sparse networks, but it tolerated only up to Byzantine nodes (where is the total network size). We answer the earlier open question by presenting fully-distributed Byzantine agreement protocols for sparse, bounded degree networks that tolerate significantly more Byzantine nodes -- up to of them.

Paper Structure

This paper contains 20 sections, 28 theorems, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model
  3. Our Contributions
  4. Prior Work and Challenges
  5. A High-level Overview of our Approach
  6. Additional Related Work
  7. Byzantine Random Walk Protocol and Theorem
  8. Definitions and Core
  9. Byzantine Random Walk Protocol
  10. Almost-Everywhere Reliable Information Dissemination
  11. Fully-Distributed Reliable Information Dissemination
  12. Low-Congestion Reliable Information Dissemination
  13. Eventual Almost-Everywhere Common Coin
  14. Eventual Almost-Everywhere Common Coin with o(n/ log n) Tolerance
  15. Detailed Primitive Description.
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $\mu = |B|/|C|$. Then, $|C|/2 \leq (1-O(\mu)) d|C|/2 \leq |E_C| \leq d |C|/2$.

Theorems & Definitions (31)

  • Definition 1: Byzantine Agreement (BA)
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 2.3: Random walk conditioned on walking in $C$
  • Lemma 2.4
  • Theorem 2.2: Byzantine Random Walk Theorem
  • Definition 2: Almost-everywhere broadcast
  • Definition 3: Almost-Everywhere to Almost-Everywhere Reliable Information Dissemination (AERID)
  • Theorem 3.1
  • ...and 21 more