Mathematical and Economic Foundations of Bitcoin

Abstract

Blockchain technology has a significant impact in multiple fields. Bitcoin was the first of its applications to rise to widespread attention. Due to its source code being publicly available (open source), it was followed quickly by a number of other cryptocurrencies as developers caught on to the opportunities it presented. The chief innovation introduced by Bitcoin was the ability to pay in a moderately secure fashion without third parties, such as banks. In this work we examine the mathematical and economic foundations upon which cryptocurrencies are built. We include a discussion of some cryptographic tools used in cryptocurrencies, such as hash functions that enable hiding information and committing to specific values; and the elliptic curve digital signature algorithm (ECDSA) used to implement Bitcoin's digital signature scheme. After covering the mathematical background we move on to the cryptocurrencies themselves. Bitcoin offers an immutable bookkeeping system based on blockchain technology, implemented using hash functions. To add transactions to the blockchain, miners calculate suitable hashes by trial and error. In order to change a transaction after it has been written onto the blockchain would require a practically infeasible amount of computation. Bitcoin transactions rely on ECDSA for authorisation. Having discussed the technical implementation of Bitcoin to some level of detail, we move on to describe two other cryptocurrencies, Zcash and Monero. These were developed to address some limitations in Bitcoin, especially when it comes to privacy considerations. We finish with a comparison of the cryptocurrencies introduced previously and a discussion of the relationship between cryptocurrencies and the society at large.