Mathematics and Data Structures in Blockchain and Ethereum

Abstract

Blockchain is a technology to exchange digital assets including cryptocurrencies, data, software, patents etc. Blockchain is designed in various types depending on different architectures and configurations for different purposes. The mathematical and cryptographic mechanisms and consensus algorithms, implemented in the Blockchain, guarantee trust, security, integrity and availability of transactions between participants. This thesis is aimed to consider mathematical foundation and algorithmic aspects of Ethereum from the base network to high level of its functionality with the comparison to mathematical mechanisms of Bitcoin. In this thesis we describe Ethereum as stack of layers. Our mission is to extract cryptographic and mathematical mechanisms implemented in each layer. We present how hash function is utilized to construct cost function and Hashcash protocol. Hashcash protocol is implemented in Proof of Work (PoW) mechanism. Also, we explain how hash function is used to connect blocks in the chain of blocks and how hash function is used to organize transactions in a Merkle tree and in a Merkle-Patricia-trie. This thesis describes how Elliptic Curve Digital Signature Algorithm (ECDSA) and Lamport signing algorithm provide integrity of transactions. Elliptic Curve Integrated Encryption Scheme (ECIES) is hybrid encryption scheme which is utilized to provide confidentiality of transactions. The Elliptic Curve Cryptography (ECC) is used to disguise identities in both Bitcoin and Ethereum network. The new approaches to provide privacy are Ring signatures and Zk-Snarks. This thesis also explains Bloom filter structure in Ethereum. The Bloom filter is a probabilistic data structure to test membership of elements in big data bases.