IEEE Blockchain Technical Briefs - Q3 2022
A collection of short technical articles

Some Mathematical Topics in Blockchain and Digital Ledger Technology

By Christopher King, Department of Mathematics, Northeastern University

Digital ledger technology (DLT) spans a vast array of topics in computer science and business, including cryptocurrencies, digital security, smart contracts, decentralized finance and numerous other applications. There are many survey and review papers which can be consulted for a broad overview of DLT and related ideas. The goal of this paper is to focus on some mathematical themes which underlie DLT, and also to indicate some newer topics which have stimulated recent mathematical analysis. One of the mathematical themes we address is the topic of one-way functions, and we show how this idea is used to provide security protocols for Bitcoin. We also describe a mathematical algorithm which is being used on digital exchanges (DEXes) to facilitate automatic trading, thereby providing profit opportunities for cryptocurrency investors. Finally we discuss some new graph-theoretic ideas related to the use of directed acyclic graphs (DAGs) in place of blockchain.

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PoW as PoS Algorithm

By Yaacov Kopeliovich

Since the invention of Nakamoto concensus there is an explosion of consensus algorithms that are divided into two main categories. The first one is PoW(Proof of Work) implemented in the first crypto asset(BTC) and the other one is PoS (Proof of Stake). PoS emerged is a concensus algorithm that emerged in a later stage due to dis-satisfication with the traditional PoW because of transaction settlement speed and environmental concerns. While these algorithms are closely related we haven't found in the literature an explanation or discussion how these algorithms related to each other. In this note we propose to carry such comparison and show that with a slight modification of PoS algorithm we can obtain a PoW like algorithm that will not require to spend energy yet it maintains all the PoW properties that make it superior in the eyes of many cryptp pundits. This note has 3 parts in the first section we explain what are PoS and PoW algorithms than in the second section we propose a PoS modification that incorporates the main features of PoW. The final section is a short philosophical discussion.

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The Mathematics Behind Blockchain

By Partho Sutra Dhor

One of the technologies that have revolutionized the world in the last decade is Blockchain. It is a public ledger or distributed database where information is verified based on the opinions of the majority of participants. This blockchain technology is used in many cases, an excellent example of which is Bitcoin. Blockchain is democratizing and decentralizing the centralized economy and information system, which is truly an unprecedented breakthrough in digital security. This article will go through the mathematical details of Blockchain technology and its future.

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The Technology and Potential of Stablecoin

By B. Ramamurthy and S. Madurai, University at Buffalo, NY

The innovation of the first working cryptocurrency Bitcoin, is two-fold: the feasibility of peer-to-peer digital money without an intermediary and the underlying enabling technology in the blockchain trust layer. Bitcoin is still running strong since its inception in 2009, fully supported by the community and without any intermediary. It is built on a robust scientific foundation of more than 50 years of research in sciences, mathematics, and algorithms. Since the introduction of Bitcoin, the blockchain and cryptocurrency ecosystem has evolved significantly. However, the main idea of Bitcoin, a peer-to-peer currency, is yet to be realized on a practical scale. The main reasons preventing ordinary people from exploring cryptocurrencies are (i) rapid advances in the technology, (ii) lack of practical information and education about blockchain and cryptocurrencies (iii) the high volatility in the prices preventing ordinary people and businesses from transacting in cryptocurrency and (iv) lack of clear policies and regulations governing this emerging area. In this brief, we discuss a technical advancement in a digital currency called stablecoin to address these concerns.

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How to Build a Blockchain: The Asynchronous Composition Model

By Partha Dey, Department of Mathematics, University of Illinois Urbana-Champaign; and Aditya Gopalan, Department of Industrial and Enterprise Systems Engineering, University of Illinois Urbana-Champaign

Inspired by blockchains, in [3], we introduce a dynamically growing model of rooted Directed Acyclic Graphs (DAGs), the asynchronous composition model, subject to random delays with finite mean. The new block at time t is connected to blocks chosen from the graph G_{max(0,t−1−ξt)} according to a blockchain rule f. Here ξ_t is the random delay at time t and the graph is updated by taking union with the graph G_t−1. This process corresponds to adding new blocks to a blockchain, where delays arise due to network communication. The main question of interest is the end structure of the asynchronous limit of the graph sequence as time increases to infinity. We consider the Nakamoto rule f_Nak, in which a vertex is uniformly selected from those furthest from the root, and the blockchain rules f₁, f₂, where in f_k a random set of k leaves is chosen without replacement. We explain that the asynchronous limits for f_Nak and f₂ are one-ended, while the asynchronous limit for f₁ is not one-ended, almost surely. We also state growth properties of the longest path for the sequence of graphs for f_Nak.