A few years ago if you asked “Can blockchains scale?” most people would give three reasons why, fundamentally, the answer is “No!”
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## The Fast Fourier Transform over finite fields

The Fast Fourier Transform (FFT) developed by Cooley and Tukey in 1965 has its origins in the work of Gauss. The FFT, its variants and extensions to finite fields, are a fundamental algorithmic tool and a beautiful example of interplay between algebra and combinatorics. There are many great resources on FFT, see ingopedia’s curated list.
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## Asynchronous Agreement on a Core Set

A challenging step in many asynchronous protocols is agreeing on a set of parties that completed some task. For example, an asynchronous protocol might start off with parties reliably broadcasting a value. Due to asynchrony and having $\leq f$ corruptions, honest parties can only wait for $n-f$ parties to complete the task. Parties may need to agree on a core set of $n-f$ such broadcasts and use them in the...
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## Can we Obtain Privacy in a Private Proof-of-Stake Blockchain? Part-II

This is Part-II of a two-part post on privacy in private proof-of-stake blockchains. In Part-I, we explored attacks on existing private PoS approaches. In this post, we will discuss some ways to obtain privacy (at the expense of safety and/or liveness).
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## Can we Obtain Privacy in a Private Proof-of-Stake Blockchain? Part-I

In this two-part post, we focus on the challenges and subtleties involved in obtaining privacy in private proof-of-stake (PoS) blockchains. For instance, designs that attempt to obtain privacy for transaction details while still relying on PoS, such as Ouroboros Crypsinous. The first part explains attacks on existing approaches, and the second part focuses on potential workarounds using differential privacy. These posts explain the intuitive ideas behind the works of Madathil...
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## The CAP Theorem and why State Machine Replication for Two Servers and One Crash Failure is Impossible in Partial Synchrony

In 1999, Fox and Brewer published a paper on the CAP principle, where they wrote:
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## $3f+1$ is needed in Partial Synchrony even against a Rollback adversary

We covered the classic DLS88 split brain impossibility result against a Byzantine adversary in a previous post:
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## Blockchains + TEEs Day 2 Summary

This is the second of the two part post on the workshop on Blockchains + TEEs that concluded last week. Here are the key ideas from Day 2. You can find the post summarizing Day 1 here.
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## Blockchains + TEEs Day 1 Summary

Our workshop on Blockchains + TEEs concluded last week. We had a fantastic series of talks and discussions on both days of the workshop. In this two part post, we highlight some key takeaways from each of the days.
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## What is the difference between PBFT, Tendermint, HotStuff, and HotStuff-2?

We recently published our work HotStuff-2 on eprint, introducing a two-phase HotStuff variant which simultaneously achieves $O(n^2)$ worst-case communication, optimistically linear communication, a two-phase commit regime within a view, and optimistic responsiveness in partially-synchronous BFT.
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## Randomization and Consensus - synchronous binary agreement for minority omission failures

Continuing the series on simple ways where randomization can help solve consensus. The model is lock-step (synchrony) with $f<n/2$ omission failures. We know that in the worst case reaching agreement takes at least $f+1$ rounds. Can randomization help reduce the expected number of rounds? In the post, we show a simple randomized consensus algorithm including a simple weak coin protocol.
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## Randomization and Consensus - synchronous binary agreement for crash failures with a perfect common coin

What is the simplest setting where randomization can help solve consensus? Assume lock-step (synchrony) with $f<n$ crash failures. We know that in the worst case reaching agreement takes at least $f+1$ rounds. This lower bound holds even if the protocol is randomized so the natural question is:
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## Can we Obtain Player Replaceability and Forensic Support Simultaneously?

Forensic support is an important property of BFT protocols that addresses the other side of security: what happens when the number of malicious parties exceeds the allowable threshold? In a previous post, we systematically studied different BFT protocols to assess their ability to detect and prove malicious behavior when safety is violated. We learned that protocols such as PBFT and HotStuff with ${\sf poly}(n)$ communication have strong forensic support, meaning...
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## What are Blockchains Useful for, Really?

Blockchains, or the decentralized ledger, are touted as the next big disruptive technology, as big as the Internet was in the 90s. What are these blockchains useful for, really? While there are relevant use cases, many examples people use that are either far too academic to be useful or are scenarios where blockchains are not the right solution in the first place. Thus, in this post, I am trying to...
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## Pairing-based Anonymous Credentials and the Power of Re-randomization

David Chaum wrote in 1985:
Large-scale automated transaction systems are imminent. The architecture chosen for these systems may have a long-term impact on the centralization of our economic system, on some of our basic liberties, and even on our democracy. The initial choice of direction will gather economic and societal momentum, making reversal increasingly less likely.
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## Player Replaceability - Towards Adaptive Security and Sub-quadratic Communication Simultaneously (Part II)

This is part II of a two-part post on player-replaceability. Part I can be found here.
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## Player Replaceability - Towards Adaptive Security and Sub-quadratic Communication Simultaneously (Part I)

This is part I of a two-part post on the concept of player-replaceability.
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## Responsiveness under omission failures

In this post, we discuss log replication responsiveness in the context of omission failures. We show how to transform the protocol in our previous post to a multi-shot version of Paxos for omission faults. The Byzantine failure case uses similar ideas and is covered in the next post of this series.
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## Set Replication - fault tolerance without total ordering

While state machine replication is the gold standard for implementing any (public) ideal functionality, its power comes at the cost of needing to totally order all transactions and as a consequence solve (Byzantine) agreement. In some cases this overhead is unnecessary.
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## What is Responsiveness?

In asynchronous protocols, latency to commit is a function of the actual maximum network delay $\delta$. In synchronous protocols, message delay is bounded by $\Delta$, and for $n/3 \leq f < n/2$, the $\Delta$ bound is used to obtain both safety and liveness. In partial synchrony, message delay is bounded by $\Delta$ after GST, and the $\Delta$ bound is used to obtain liveness.
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