By Vitalik Buterin, PhD at University of Basel A common strand of mathematics argues that, rather than being one single kind of infinity, there are actually an infinite hierarchy of different levels of infinity. Whereas the size of the set of integers is just plain infinite, and the set of…

By Vitalik Buterin and Glen Weyl Wealthy societies around the world are facing a growing crisis of confidence in established authorities. Stagnating economies, mounting inequality, political corruption and the increasing monopolization of technology for the benefit of elites have provoked a populist backlash. We share and are driven by these…

Traditional consensus algorithms, whether they operate in a synchronous, partially asynchronous or fully asynchronous network model, and whether they are designed around simple faults, Byzantine faults or accountable faults, generally operate in a model where there is a fixed set of participants in the protocol, with the assumption that at…

Last week Yoichi released a blog post detailing the process of formally proving safety and liveness properties of my “minimal slashing conditions”. This is a key component of a Byzantine-fault-tolerant, safe-under-asynchrony and cryptoeconomically safe consensus algorithm that is at the core of my proof of stake roadmap. In this post…

“Decentralization” is one of the words that is used in the cryptoeconomics space the most frequently, and is often even viewed as a blockchain’s entire raison d’être, but it is also one of the words that is perhaps defined the most poorly. Thousands of hours of research, and billions of…

This is the third part of a series of articles explaining how the technology behind zk-SNARKs works; the previous articles on quadratic arithmetic programs and elliptic curve pairings are required reading, and this article will assume knowledge of both concepts. Basic knowledge of what zk-SNARKs are and what they do…

Trigger warning: math. One of the key cryptographic primitives behind various constructions, including deterministic threshold signatures, zk-SNARKs and other simpler forms of zero-knowledge proofs is the elliptic curve pairing. Elliptic curve pairings (or “bilinear maps”) are a recent addition to a 30-year-long history of using elliptic curves for cryptographic applications…

As Casper continues to reach an increasingly stabilized form, there has been increased interest in the various parameters that are going to be set in the protocol, including the interest rate, fees, withdrawal period, speed, the minimum amount of ETH needed to stake, etc. This article will be the first…

Systems like Ethereum (and Bitcoin, and NXT, and Bitshares, etc) are a fundamentally new class of cryptoeconomic organisms — decentralized, jurisdictionless entities that exist entirely in cyberspace, maintained by a combination of cryptography, economics and social consensus. They are kind of like BitTorrent, but they are also not like BitTorrent…