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Mathematical Consideration of Pyramid Schemes
Since the birth of Bitcoin, countless digital assets have been created one after another on the blockchain. By using these digital assets, cross-border remittances have become significantly easier than before. Rise in Scam Alongside Digital Asset Adoption While these developments have brought convenience, cross-border crimes involving digital assets (especially scam) are frequently occurring. Examples include…
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Consensus Algorithm in Bitcoin Part 2
This section explains why Bitcoin’s consensus algorithm functions effectively in practice, considering potential attackers (malicious nodes). Consensus Formation Through the Longest Chain Rule In the Bitcoin blockchain⛓️, miners, including malicious ones, autonomously extend the chain. Inevitably, chain forks occur, but Bitcoin enforces a rule that only the longest chain is considered valid*1. This rule allows…
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Consensus Algorithm in Bitcoin Part 1
Distributed Network Systems and the Byzantine Generals Problem In distributed network systems, achieving consensus involves addressing the Byzantine Generals Problem. It has been shown in the context of Decentralized Consensus Formation and the Byzantine Generals Problem that if more than one-third of the nodes are malicious, reaching consensus is impossible. Even when the Byzantine Generals…
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Decentralized Consensus Formation and the Byzantine Generals Problem
Overview of the Byzantine Generals Problem A well-known issue in decentralized network systems is the Byzantine Generals Problem (Byzantine Fault). This problem addresses consensus formation in decentralized network systems and was formalized in 1982 by Leslie Lamport and his colleagues: Leslie Lamport, Robert Shostak, and Marshall Pease, The Byzantine Generals Problem, SRI International, 1982. Leslie…
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Representation Theory is Mathematics
The terms “Expression” and “Representation” are frequently used in mathematics and everyday language, both carrying the broad meaning of “indicating or describing something.” However, their usage and nuances differ. In this article, we will clarify the distinctions between these two concepts and explain why the field of mathematics known as “表現論” in Japanese is called…