Synchronous consensus protocols can tolerate $f < n/2$ Byzantine failures but for $n/3 <f <n/2$ must depend on the maximum network delay $\Delta$ for their safety and progress. So these protocols must set $\Delta$ to be much larger than the actual network delay $\delta \ll \Delta$. The good news is in the multi-shot (blockchain) scenarios, modern synchronous protocols such as Sync HotStuff can essentially pipeline the $\Delta$-dependent delay:
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## Consensus cheat sheet

Crash
Omission
Byzantine
Synchrony
$f<n$ is possible $f+1$ round executions must exist
$f \geq n/2$ is impossible
$f<n/2$ possible with PKI / PoW $f \geq n/3$ impossible without PKI/PoW
Partial Synchrony
$f \geq n/2$ is impossible
$f<n/2$ is possible
$f<n/3$ is possible $f \geq n/3$ is impossible
Asynchrony
non terminating executions must exist
$f<n/2$ possible in $O(1)$ expected
$f<n/3$ possible in $O(1)$ expected
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## The Ideal State Machine Model: Multiple Clients and Linearizability

We introduced state machines and state machine replication in an earlier post. In this post, we elaborate on the exact notion of safety and liveness that can be obtained by an ideal state machine when there are multiple clients.
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## Nakamoto's Longest-Chain Wins Protocol

In this post, we will cover Nakamoto’s Consensus protocol presented in the Bitcoin whitepaper. There are a lot of posts and videos that explain this seminal and simple protocol. Our goal in this post will be to intuitively piece out the need for different aspects of the protocol, such as proof-of-work and how network synchrony plays a role in the protocol.
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## Crusader Agreement with $\leq 1/3$ Error is Impossible for $n\leq 3f$ if the Adversary can Simulate

The classic FLM lower bound says that in Synchrony, Byzantine Agreement is impossible when $n \leq 3f$. We discussed this important bound in a previous post. In this post we strengthen the FLM lower bound in two important ways: Maybe randomization allows circumventing the FLM lower bound? No! Even allowing $\leq 1/3$ error, this bound still holds (based on unpublished work of Yao and Karlin, see the work of Graham...
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## Distributed consensus made simple (for real this time!)

Multi-Paxos is the de facto solution for deciding a log of commands to execute on a replicated state machine, yet it’s famously difficult to understand, motivating the switch to ‘simpler’ consensus protocols such as Raft. The conventional wisdom is that the best way to use Paxos (aka Synod, or single-shot Paxos), to decide a log of commands is to run many instances of it, where the $i^{th}$ instance decides the...
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## The round complexity of Reliable Broadcast

Reliable Broadcast is an important building block of many Asynchronous protocols. There is a broadcaster that has some input value, $v$, and a non-faulty party that terminates needs to output a value. Reliable Broadcast is defined via two properties:
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## Optimal Communication Complexity of Authenticated Byzantine Agreement

Communication complexity of Byzantine Agreement (BA) has been studied for decades. Dolev and Reischuk showed that quadratic communication is necessary for BA with perfect security, i.e., without error. Berman, Garay, and Perry showed that this lower bound is tight for the unauthenticated setting (without signatures) with $f < n/3$. However, for $f \ge n/3$, the best solution so far is still the Dolev-Strong protocol with cubic communication, so a linear...
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## Information Theoretic HotStuff (IT-HS): Part One

This post is the first of two on Information Theoretic HotStuff (IT-HS). Information Theoretic HotStuff is a Byzantine Consensus protocol in the partially synchronous model. It replaces all of HotStuff’s cryptographic signatures with simple information theoretic message passing techniques over authenticated channels. Information theoretic protocols are often easy to reason about, form a great introduction for learning the basics of consensus, have less of an attack surface, and highlight useful...
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## The Private Set Intersection (PSI) Protocol of the Apple CSAM Detection System

In this post, we will discuss the cryptographic construction used in Apple’s new system for detecting CSAM - Child Sexual Abuse Material. This cryptographic construction implements a new variant of PSI - Private Set Intersection. While a lot of attention has been paid to the broad implications of this system, the technical details of the PSI construction have not been highlighted, even though Apple published a detailed technical description of...
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## Simplifying Raft with Chaining

Raft is a consensus algorithm for deciding a sequence of commands to execute on a replicated state machine. Raft is famed for its understandability (relative to other consensus algorithms such as Paxos) yet some aspects of the protocol still require careful treatment. For instance, determining when it is safe for a leader to commit commands from previous leaders or when it is safe for servers to delete or overwrite commands...
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## Neither Non-equivocation nor Transferability alone is enough for tolerating minority corruptions in asynchrony

In this post, we explore a theorem of Clement, Junqueira, Kate, and Rodrigues from PODC 2012 regarding the limits of non-equivocation. Informally, this theorem says that neither Non-equivocation nor Transferability alone is enough for tolerating minority corruptions in asynchrony.
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## Benign Hotstuff

In this post we describe a variant of Paxos (or Raft or Lock-Commit) that is inspired by looking through the lens of HotStuff and Blockchain protocols. The most noticeable difference is that while Paxos and Raft aim to maintain a stable Primary/Leader (and change views infrequently), in Benign Hotstuff (as in HotStuff), the Primary is rotated every round! A more subtle difference is that in Paxos and Raft each block...
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## Living with Asynchrony: the Gather protocol

A very useful tool in Asynchronus distributed computing is Reliable Broadcast, or simply called Broadcast. It allows a leader to send a message, knowing that all parties will eventually receive the same message, even if a malicious adversary control $f$ parties and $f<n/3$. Broadcast is deterministic and takes just a constant number of rounds.
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## Good-case Latency of Byzantine Broadcast: the Synchronous Case

In our first post, we presented a summary of our good-case latency results for Byzantine broadcast (BB) and state machine replication (SMR), where the good case measures the latency to commit given that the broadcaster or leader is honest. In our second post, we discussed our results for partial synchrony, and described a new BFT SMR protocol named (5f-1)-SMR that can commit a decision within $2$ rounds in the good...
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## 2-round BFT SMR with n=4, f=1

Guest post by Zhuolun Xiang
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## Good-case Latency of Byzantine Broadcast: a Complete Categorization

Guest post by Zhuolun Xiang
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## What is a Merkle Tree?

In this post, we will demystify Merkle trees using three examples of problems they solve:
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## Raft does not Guarantee Liveness in the face of Network Faults

Last month, Cloudflare published a postmortem of a recent 6-hour outage caused by a partial switch failure which left etcd unavailable as it was unable to establish a stable leader. This outage has understandably led to discussion online about exactly what liveness guarantees are provided by the Raft consensus algorithm in the face of network failures.
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## A Simple and Succinct Zero Knowledge Proof

There is a popular belief that succinct proofs and zero-knowledge proofs are a type of moon math. In this post, our goal is to present a simple proof system that can provide an introduction and intuition to this space. Perhaps surprisingly, the only tool we will use is the Theorem that non-trivial degree-at-most-$d$ polynomials over a field have at most $d$ roots.
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