This page contains material for a graduate course on Blockchains and Distributed Computing with a dash of Cryptography (stay tuned for more). Or read posts chronologically.

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Basics, Foundations, and Classics

Start with the definition of Consensus and Agreement. Then learn about the network model, the threshold adversary model, and the power of the adversary. Many protocols need a trusted setup phase. The consensus cheat sheet gives a quick overview of what is possible and impossible. You can build half a course just from the upper bounds and lower bounds linked from it.

Variants of Consensus and Broadcast

Approximate agreement is a variation that considers rational input values.

This post covers several relaxations of Broadcast.

Synchronous Protocols

The Synchronous model is a good place to start because protocols are simpler.

• Classic Byzantine agreement protocol for $f\leq n/2$: the Phase-King.

• Classic Byzantine Broadcast protocol (with a PKI) for $f < n$: the Dolev-Strong Authenticated Broadcast protocol.

• A non-equivocation protocol (with a PKI) for $f < n$: Crusader Broadcast.

More recent State Machine Replication protocols:

• Sync HotStuff.
• An optimal optimistically responsive synchronous protocol.
• A simple, streamlined synchronous protocol called Streamlet.
• Survey of authenticated protocols under the synchrony assumption.

Use of randomization in synchronous agreement protocols:

• For crash failures.
• For omission failures.

Partially Synchronous Protocols

Partial synchrony is one of the most used models in real word systems today.

Single shot Paxos.

Taking ideas from the Byzantine setting and applying them to the benign setting:

• Benign Hotstuff.
• Simplifing Raft with Chaining.
• Log Paxos is a modern take on multi-Paxos.

The Byzantine setting:

• An important building block: provable broadcast.
• Single shot PBFT.
• Two Round HotStuff with SMR and accountability.
• Single shot to SMR is covered here.
• Information Theoretic HotStuff.

Asynchronous Protocols

Fundamental lower bound in asynchrony: The FLP lower bound showing the impossibility of consensus under even one crash fault.

Important building blocks in asynchrony:

1. The Reliable Broadcast protocol.
2. The Reliable Gather is a multi-leader generalization of reliable broadcast.
3. We discusses how to measure round complexity in asynchrony and also improved round complexity for reliable broadcast.

Series on Asynchronous Agreement:

1. Define the problem;
2. present Ben-Or’s protocol;
3. provide a modern version;
4. Introduce Crusader Agreement and Binding Crusader Agreement;
5. use BCA to efficiently solve Binary Byzantine Agreement from a strong common coin.

Reaching Asynchronous Agreement on a Core Set.

State Machine Replication

Define state machine replication (SMR).

Levels of SMR fault tolerance.

The ideal state machine model and Linearizability.

The scalability and performance of a State Machine Replication system are not just about consensus, but also about data and execution.

How DAGs improve the performance of SMR.

To learn about upper bounds, start with a simple SMR for crash failures. Then extend SMR to omission failures: First via single shot and then via the lock-commit paradigm to multi-shot consensus.

Lower Bounds

Lower bounds give us powerful tools to understand the fundamental limitations and model assumptions.

Lower bounds on the size of the adversary:

• Folklore (aka CAP theorem for synchrony): Consensus with Omission failures requires $f<n/2$.

• The CAP theorem for partial synchrony: Any system, in the face of a perceived 50-50 split, can either maintain safety or liveness, but not both.

• Split brain attacks:
  • Dwork, Lynch, Stockmeyer 1988 (DLS) lower bound: Byzantine Consensus in Partial Synchrony requires $f<n/3$.
  • CJKR lower bound: Neither Non-equivocation nor Transferability alone is enough for tolerating minority corruptions in asynchrony.
  • TEEs without state need $3f+1$ in Partial Synchrony because of the rollback adversary.

• Fischer, Lynch, Merritt (FLM) lower bound: Byzantine Consensus with no PKI (or more generally when the adversary can simulate) requires $f<n/3$.

• Strengthening the FLM lower bound with randomization and a weaker problem: Crusader Agreement with $\leq 1/3$ error is impossible for $n\leq 3f$ if the adversary can simulate.

Lower bounds on the number of messages

• Dolev and Reischuk 1982 (DR) lower bound: Consensus (even with omission failures) needs a quadratic number of messages.
• Extending Dolev and Reischuk to Byzantine crusader broadcast.

Lower bounds on the round complexity

• Fischer, Lynch, Paterson 1985 (FLP) lower bound, three part series:
  • Consensus must have some initial state that is uncommitted:
  • This implies executions with at least $f+1$ rounds in Synchrony.
  • The FLP result: non-terminating executions in Asynchrony.

Lower bounds due to privacy requirements

• Ben-Or, Kelmer, Rabin 1994 (BKR) lower bound: Asynchronous Verifiable Secret Sharing must have a non-zero probability of not terminating.

Liveness attacks

• Raft does not guarantee liveness under omission faults.

Blockchains

What is a blockchain?

What are blockchains useful for, really?

What was the first blockchain (or how to timestamp a digital document)?

What is Nakamoto Consensus? How do you prove Nakamoto Consensus is secure?

Do proof-of-work blockchains need any setup assumptions??

What does checkpointing a blockchain mean?

What is the problem of selfish mining?

The simplest L2 solution is a payment channel.

What is player replaceability? A series:

• Defining the problem and achieving a committee-based sub-quadratic communication protocol
• Achieving adaptive security for the committee-based protocol

Cryptography

• Cryptographic hash functions.

• Merkle trees.

• A glimpse at Zero knowledge proofs

• Bilinear accumulators from polynomial commitments.

• range proofs from polynomial commitments.

• Private set intersection: part 1 and part 2. Apple is using PSI for CSAM detection.

Polynomials

• A lightweight mathematical intro to Polynomials over a finite field.

• The basics of Polynomial secret sharing.

• The Fast Fourier Transform over finite fields.

Secret Sharing and MPC

Polynomial secret sharing is a base for deep connections between cryptography and distributed computing.

• Polynomial secret sharing against a passive adversary

• Against a crash adversary.

• The BGW88 protocol for Verifiable Secret Sharing.

• Chaum’s Dining Cryptographers and the additivity of polynomial secret sharing.

Research Oriented Posts

• What is the difference between PBFT, Tendermint, SBFT, and HotStuff ?

• Survey of modern Authenticated Synchronous BFT protocols (updated in March 2021).

• Sync HotStuff.

• Optimal Optimistic Responsivness.

• Encrypted Blockchain Databases part 1 and part 2.

• Asynchronous Fault-Tolerant Computation with Optimal Resilience.

• Resolving the Availability-Finality Dilemma.

• BFT Protocol Forensics.

• Can we Obtain Player Replaceability and Forensic Support Simultaneously?

• Good-case Latency of Byzantine Broadcast: the Synchronous Case and a Complete Categorization.

• 2-round BFT SMR with n=4, f=1.

• Optimal Communication Complexity of Authenticated Byzantine Agreement

• Good-case Latency of Rotating Leader Synchronous BFT

• EIP-1559 in Retrospect

• Colordag: From always-almost to almost-always 50% selfish mining resilience

• Consensus by Dfinity: in synchrony and Consensus by Dfinity - Part II (Internet Computer Consensus): in partial synchrony

• Sandglass: Safe Permissionless Consensus.

• DAG Meets BFT - The Next Generation of BFT Consensus.

• He-HTLC - Revisiting Incentives in HTLC.