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COOL Is Optimal in Error-Free Asynchronous Byzantine Agreement

Jinyuan Chen

TL;DR

OciorACOOL addresses asynchronous Byzantine agreement with information-theoretic security by extending COOL through a carefully integrated set of primitives: UA-based progression, asynchronous BA (ABBA) and BRBA, and online error correction. The protocol achieves error-free BA with total communication $O(\\max\\{n\\ell, nt\\log q\\})$ bits in $O(1)$ rounds, using a single asynchronous binary-BA invocation under the optimal resilience $n\\ge 3t+1$, while retaining the classic $(n, k)$ ECC with $k=t/3$. Its construction combines COOL-UA[1], COOL-UA[2], ABBA, BRBA, and COOL-HMDM[2], together with an analysis that proves Consistency, Validity, and Termination, anchored by a key lemma that bounds the number of honest input groups. Compared to list-decoding approaches, OciorACOOL delivers a practical, information-theoretic asynchronous BA with substantial communication savings and without relying on cryptographic primitives, enabling robust distributed consensus in asynchronous environments.

Abstract

COOL (Chen'21) is an error-free, information-theoretically secure Byzantine agreement (BA) protocol proven to achieve BA consensus in the synchronous setting for an $\ell$-bit message, with a total communication complexity of $O(\max\{n\ell, nt \log q\})$ bits, four communication rounds in the worst case, and a single invocation of a binary BA, under the optimal resilience assumption $n \geq 3t + 1$ in a network of $n$ nodes, where up to $t$ nodes may behave dishonestly. Here, $q$ denotes the alphabet size of the error correction code used in the protocol. In this work, we present an adaptive variant of COOL, called OciorACOOL, which achieves error-free, information-theoretically secure BA consensus in the asynchronous setting with total $O(\max\{n\ell, n t \log q\})$ communication bits, $O(1)$ rounds, and a single invocation of an asynchronous binary BA protocol, still under the optimal resilience assumption $n \geq 3t + 1$. Moreover, OciorACOOL retains the same low-complexity, traditional $(n, k)$ error-correction encoding and decoding as COOL, with $k=t/3$.

COOL Is Optimal in Error-Free Asynchronous Byzantine Agreement

TL;DR

OciorACOOL addresses asynchronous Byzantine agreement with information-theoretic security by extending COOL through a carefully integrated set of primitives: UA-based progression, asynchronous BA (ABBA) and BRBA, and online error correction. The protocol achieves error-free BA with total communication bits in rounds, using a single asynchronous binary-BA invocation under the optimal resilience , while retaining the classic ECC with . Its construction combines COOL-UA[1], COOL-UA[2], ABBA, BRBA, and COOL-HMDM[2], together with an analysis that proves Consistency, Validity, and Termination, anchored by a key lemma that bounds the number of honest input groups. Compared to list-decoding approaches, OciorACOOL delivers a practical, information-theoretic asynchronous BA with substantial communication savings and without relying on cryptographic primitives, enabling robust distributed consensus in asynchronous environments.

Abstract

COOL (Chen'21) is an error-free, information-theoretically secure Byzantine agreement (BA) protocol proven to achieve BA consensus in the synchronous setting for an -bit message, with a total communication complexity of bits, four communication rounds in the worst case, and a single invocation of a binary BA, under the optimal resilience assumption in a network of nodes, where up to nodes may behave dishonestly. Here, denotes the alphabet size of the error correction code used in the protocol. In this work, we present an adaptive variant of COOL, called OciorACOOL, which achieves error-free, information-theoretically secure BA consensus in the asynchronous setting with total communication bits, rounds, and a single invocation of an asynchronous binary BA protocol, still under the optimal resilience assumption . Moreover, OciorACOOL retains the same low-complexity, traditional error-correction encoding and decoding as COOL, with .

Paper Structure

This paper contains 16 sections, 17 theorems, 78 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Primitives
  3. $\mathsf{OciorACOOL}$
  4. Revisiting $\mathsf{COOL}$ for the Synchronous Setting
  5. $\mathsf{COOL}\text{-}\mathsf{UA}$
  6. Binary $\mathsf{BA}$
  7. $\mathsf{COOL}\text{-}\mathsf{HMDM}$
  8. The Challenge of $\mathsf{COOL}$ in the Asynchronous Setting
  9. Overview of $\mathsf{OciorACOOL}$ for asynchronous stetting
  10. $\mathsf{COOL}\text{-}\mathsf{UA}[1]$
  11. Set the Input Value $\tilde{\boldsymbol{w}}_i$ for $\mathsf{COOL}\text{-}\mathsf{UA}[2]$
  12. $\mathsf{COOL}\text{-}\mathsf{UA}[2]$
  13. $\mathsf{ABBA}$ and $\mathsf{BRBA}$
  14. $\mathsf{COOL}\text{-}\mathsf{HMDM}[2]$
  15. Analysis of $\mathsf{OciorACOOL}$
  16. ...and 1 more sections

Key Result

Theorem 1

In $\mathsf{OciorACOOL}$, given $n\geq 3t+1$, if one honest node outputs a value $\boldsymbol{w}^{\star}$, then every honest node eventually outputs a value $\boldsymbol{w}^{\star}$, for some $\boldsymbol{w}^{\star}$.

Figures (2)

  • Figure 1: A block diagram of the $\mathsf{COOL}$ protocol, which consists of the $\mathsf{COOL}\text{-}\mathsf{UA}$, binary $\mathsf{BA}$, and $\mathsf{COOL}\text{-}\mathsf{HMDM}$ algorithms.
  • Figure 2: A block diagram of the $\mathsf{OciorACOOL}$ protocol, which consists of the $\mathsf{COOL}\text{-}\mathsf{UA}[1]$, $\mathsf{COOL}\text{-}\mathsf{UA}[2]$, asynchronous binary $\mathsf{BA}$ ($\mathsf{ABBA}$), binary reliable Byzantine agreement ($\mathsf{BRBA}$), and $\mathsf{COOL}\text{-}\mathsf{HMDM}[2]$ algorithms. If an invoked $\mathsf{ABBA}$ algorithm already guarantees the Totality property, then $\mathsf{BRBA}$ is not required in $\mathsf{OciorACOOL}$.

Theorems & Definitions (37)

  • Definition 1: Byzantine Agreement ($\mathsf{BA}$)
  • Definition 2: Reliable Byzantine Agreement ($\mathsf{RBA}$)
  • Definition 3: Reliable Broadcast ($\mathrm{RBC}$)
  • Definition 4: Honest-Majority Distributed Multicast ($\mathrm{HMDM}$) Chen:2020arxivChenDISC:21ChenOciorCOOL:24
  • Definition 5: Strongly-Honest-Majority Distributed Multicast ($\mathsf{SHMDM}$) ChenOciorMVBA:24
  • Definition 6: Unique Agreement ($\mathrm{UA}$) Chen:2020arxivChenDISC:21ChenOciorCOOL:24
  • Theorem 1: Consistency
  • proof
  • Theorem 2: Validity
  • proof
  • ...and 27 more