arxiv: 2605.10372 · v2 · submitted 2026-05-11 · 💻 cs.DC

Recognition: 1 theorem link

· Lean Theorem

Amortized Asynchronous Byzantine Reliable Broadcast with Optimal Resilience

Alvin Hong Yao Yan, Jialin Li, Michael Yiqing Hu

Authors on Pith no claims yet

Pith reviewed 2026-05-13 07:21 UTC · model grok-4.3

classification 💻 cs.DC

keywords Byzantine reliable broadcastasynchronous networksamortized communicationoptimal resiliencemulti-shot protocolscommunication complexitydistributed computing

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The pith

Amortized multi-shot protocol achieves asymptotically optimal O(n|m|) communication for asynchronous Byzantine reliable broadcast while preserving f < n/3 resilience.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to apply amortization to Byzantine reliable broadcast in fully asynchronous networks by running multiple instances together. Initial rounds establish incremental guarantees that allow every later instance to finish after only one extra round. This produces linear message complexity per broadcast once messages are large enough, without sacrificing the standard optimal resilience bound or introducing randomness. A reader would care because earlier sub-quadratic solutions for the asynchronous case usually required either weaker fault tolerance or probabilistic guarantees.

Core claim

The protocol structures BRB across multiple rounds that each add incremental guarantees; once the initial rounds complete, each subsequent BRB instance requires only a single additional round. This amortization yields asymptotic O(n|m|) message complexity for sufficiently large messages, Ω(n) round complexity in the worst case, and Ω(1) round complexity along optimistic delivery paths, all while keeping optimal resilience against fewer than n/3 Byzantine faults.

What carries the argument

The amortization strategy that supplies incremental additive guarantees across structured rounds, reducing each new BRB instance to one extra round after setup.

If this is right

After the initial rounds, each additional broadcast costs only O(n) messages per node for large |m|.
The protocol functions correctly under the standard asynchronous message-passing model with no timing assumptions.
Optimal resilience f < n/3 is retained without probabilistic techniques or weaker fault bounds.
Round complexity stays linear in the worst case but reaches constant under favorable network conditions via the optimistic path.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same incremental-guarantee pattern could be applied to other multi-shot primitives such as reliable broadcast variants or atomic broadcast.
Performance measurements across a range of message sizes would identify the practical threshold where amortization begins to dominate.
Hybrid protocols could monitor network behavior and fall back to the optimistic single-round path when delays remain bounded.
The approach suggests that amortization may reduce communication costs in other asynchronous agreement tasks without changing resilience.

Load-bearing premise

Messages must be large enough for the per-instance communication savings to outweigh the fixed cost of the initial setup rounds.

What would settle it

A concrete asynchronous execution with f = floor((n-1)/3) faults in which the total messages for k broadcasts exceed O(k n |m|) or in which some correct node fails to deliver a message.

read the original abstract

Byzantine Reliable Broadcast (BRB) is a fundamental primitive in distributed computing and cryptographic systems. Reducing the communication complexity of BRB protocols remains an important research direction. However, most work focuses on synchronous networks, with limited attention to the more challenging setting of network \textit{asynchrony}. Achieving sub-quadratic communication for asynchronous BRB typically requires probabilistic approaches that sacrifice optimal $f=\frac{n}{3}$ resilience. In this work, we present a multi-shot BRB algorithm for asynchronous networks that maintains optimal resilience through an underutilized technique: \textit{amortization}. Our protocol structures BRB across multiple rounds, where each round provides incremental additive guarantees. Once these initial rounds complete, each subsequent BRB instance requires only a single additional round. This amortization strategy achieves asymptotic optimal $O(n|m|)$ message complexity when messages are sufficiently large, with $\Omega(n)$ round complexity in the worst case. Under favorable conditions, an optimistic delivery path reduces the round complexity to $\Omega(1)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a multi-shot async BRB that keeps optimal f < n/3 resilience by amortizing across instances to reach O(n|m|) communication for large messages.

read the letter

The main point is that this work presents a multi-shot asynchronous Byzantine reliable broadcast protocol that achieves optimal resilience of f < n/3 through amortization across instances. It structures the broadcasts so that some initial rounds build up the necessary guarantees, after which each new instance can complete in a single round under favorable conditions. This leads to O(n |m|) message complexity overall when messages are large, which is asymptotically optimal. The worst-case round complexity stays linear in n, but the optimistic path aims for constant rounds. What the paper does well is identify the gap in async BRB work and apply amortization in a way that avoids dropping resilience or going probabilistic. The abstract lays out the complexity claims directly. The potential soft spot is in the single-round delivery mechanism. Standard async BRB needs several phases to ensure intersecting quorums and prevent Byzantine nodes from causing disagreement. The paper relies on the initial rounds to pre-establish state, but it will be important to verify that this state holds up against asynchrony and faults without allowing inconsistent deliveries. The stress-test concern about agreement is reasonable to check against the full protocol description. This paper is for researchers working on efficient distributed primitives in asynchronous settings. It should get a serious referee because the claims are specific and the approach is a fresh angle on a known hard problem. I recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a multi-shot Byzantine Reliable Broadcast (BRB) protocol for asynchronous networks that uses an amortization technique across multiple rounds to achieve optimal resilience (f < n/3) and O(n|m|) message complexity for sufficiently large messages, with worst-case round complexity Ω(n) but optimistic path reducing to Ω(1).

Significance. If the protocol and its proofs are correct, this result would be significant for distributed systems, as it provides a deterministic solution with optimal resilience and amortized optimal communication in the challenging asynchronous model, improving upon prior work that either requires higher communication or uses probabilistic methods with reduced resilience.

major comments (2)

[Protocol Description] The description of the optimistic single-round delivery path for subsequent BRB instances (in the protocol section following the initial rounds): the claim that each subsequent instance requires only one additional round while preserving agreement appears to lack a described mechanism for collecting intersecting quorums of size 2f+1. Standard asynchronous BRB constructions require multiple sequential phases to prevent a Byzantine node from delivering inconsistent values to different correct nodes; without explicit cross-verification or pre-established shared state that survives asynchrony, this risks violating the agreement property.
[Complexity Analysis] § on complexity analysis: the asymptotic O(n|m|) message complexity is claimed to be achieved via amortization when messages are large, but the analysis does not explicitly bound the initial setup cost or show how it is amortized over a concrete number of instances to dominate and yield the stated bound (including any dependence on the number of BRB instances).

minor comments (2)

[Abstract] The abstract refers to 'incremental additive guarantees' without a brief definition or forward reference; adding one sentence of clarification would improve readability.
[Preliminaries] Notation for message size |m| is used without an explicit definition in the preliminaries; a short reminder of the model parameters (n, f, message size) would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript describing the amortized asynchronous Byzantine Reliable Broadcast protocol. We address each major comment below and have revised the manuscript to improve clarity and completeness.

read point-by-point responses

Referee: [Protocol Description] The description of the optimistic single-round delivery path for subsequent BRB instances (in the protocol section following the initial rounds): the claim that each subsequent instance requires only one additional round while preserving agreement appears to lack a described mechanism for collecting intersecting quorums of size 2f+1. Standard asynchronous BRB constructions require multiple sequential phases to prevent a Byzantine node from delivering inconsistent values to different correct nodes; without explicit cross-verification or pre-established shared state that survives asynchrony, this risks violating the agreement property.

Authors: We thank the referee for this observation. The amortization technique in our protocol establishes persistent shared state across the initial rounds, consisting of verified message deliveries and intersecting quorums of size 2f+1 that are guaranteed by the reliable broadcast properties. This state survives asynchrony and is maintained locally by correct nodes, enabling subsequent instances to perform delivery in a single round by cross-verifying against the pre-established quorums without requiring additional phases. We have revised the protocol description section to explicitly detail this mechanism, including how the shared state is updated and used for verification in the optimistic path. revision: yes
Referee: [Complexity Analysis] § on complexity analysis: the asymptotic O(n|m|) message complexity is claimed to be achieved via amortization when messages are large, but the analysis does not explicitly bound the initial setup cost or show how it is amortized over a concrete number of instances to dominate and yield the stated bound (including any dependence on the number of BRB instances).

Authors: The referee is correct that the complexity analysis requires a more explicit accounting of setup costs. We have revised the relevant section to bound the initial setup at O(n^2) messages (independent of |m|). This cost is amortized over k instances, yielding a per-instance cost of O(n^2/k + n|m|). For sufficiently large k and |m| (specifically when |m| = Ω(n)), the bound simplifies to O(n|m|). The revised analysis now states the dependence on k and the precise conditions for the asymptotic claim. revision: yes

Circularity Check

0 steps flagged

No circularity: new protocol construction with independent complexity analysis

full rationale

The paper introduces a multi-shot BRB protocol using amortization across rounds to achieve O(n|m|) message complexity under standard asynchronous model assumptions (n nodes, f < n/3 faults). The central claims follow from the described structure (initial rounds for guarantees, then single-round instances) without reducing to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. No equations or prior-author uniqueness theorems are invoked in a way that collapses the result to its inputs. This is a standard constructive result in distributed computing.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities beyond standard distributed computing assumptions; full paper would be needed to audit.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Our protocol structures BRB across multiple rounds, where each round provides incremental additive guarantees. Once these initial rounds complete, each subsequent BRB instance requires only a single additional round.

Reference graph

Works this paper leans on

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