This book provides a foundation for the mathematics of blockchain. It is inspired by a general analysis and synthesis of the current knowledge of blockchain technology and starts by laying a foundation for the mathematics of blockchain. The aim is for research in the area of blockchain to lead their study from the construction highlighted in this book.
First, the basis of a blockchain is set: block, transaction contents, block header, Merkle tree, nonce, Proof-of-Work. Connections with elliptic curves and cryptographic signatures are made.
Second, the book ties this with a Graph and Matrix Theories approach and models the peer-to-peer relationship through the Bitcoin Network.
Third, it is proposed further modelling, notably around halving, optimal storing, or diffusion of information, which are consequences of the mathematical foundation. The notion of Entropy of Privacy and the Particles model are introduced.Finally, the mathematical statements therein are proven and essential reminders can be found before each section, so the content can be accessible from a graduate level.
Author(s): Julien Riposo
Publisher: Springer
Year: 2023
Language: English
Pages: 154
City: Cham
Preface
Contents
Chapter 1: Motivation for Mathematics of Blockchain
Chapter 2: Some Fundamentals of Mathematics of Blockchain
1 Preliminaries on Analysis
1.1 Basic Functions and Conventions
1.2 Hashing Function
1.3 Probability
1.4 Concatenation
1.5 Memoryless Property of Positive Continuous Random Variables
2 Block Modelling
2.1 Mathematics of Blockchain
2.2 Time for Mining Blocks
3 Merkle Tree
3.1 Complete Merkle Tree
3.2 Uncomplete Merkle Tree
4 Transaction Data
Chapter 3: Digital Signature
1 Preliminaries on Needed Algebra
1.1 Group
1.2 Field
1.3 Morphism
1.4 Characteristic
2 Elliptic Curves
2.1 Weierstrass Equation
2.2 Non-singularity
3 The Underlying Abelian Group
4 Elliptic Curve Digital Signature Algorithm (ECDSA) and Schnorr Signature
5 Back to the Blockchain
Chapter 4: Blockchain Contributors: The Network of Users
1 Preliminaries on Graph and Matrix Theories
1.1 Notations and Facts
1.2 Equivalence Between the Set of Graphs and the Set of Matrices
2 Bitcoin Network
3 Some Modeling for the Network of Users
3.1 Perron-Frobenius Theorem
3.2 Invariants of an Adjacency Matrix
3.3 Random Walk and Principal Invariants
3.4 Symmetric Group and Principal Invariants
3.5 First Hitting Times and Absorption Probability
3.6 A Quick Word on Diffusion and Propagation of Information
4 Diffusion Models on a Graph
4.1 ``S´´ Shapes
4.2 Bass Model
4.3 DeGroot Model
4.4 Diffusion on the Peer-to-Peer Network
Chapter 5: Halving and Cycles Theorem
1 Preliminaries on Modular Arithmetic
2 Bitcoin Halving
3 The Digit Pattern
4 The Cycles Theorem
4.1 Observation of Some Cycles
4.2 The Cycles Theorem
4.3 Binary Tree Representation of Halving
4.4 On an Algorithm for Predicting the Satoshi Digits
Chapter 6: On Improving the Merkle Trees: The n-Trees
1 Description of an n-Tree
2 Developments on an n-Tree Implementation
3 Empirical Comparison of Computation Times
4 A Mathematical Challenge for Finding the Right n
4.1 Problem Position
4.2 Partial Solution
5 A Full Solution to the Mathematical Challenge for Finding the Right n
5.1 The Cost Function
5.2 Considering the n-Tree Nodes in the Cost Function
5.3 Empirical Analysis
5.4 A Word on an Analytical Derivation of the Optimal Value
6 The Dynamic n-Tree
6.1 Principles
6.2 Static Tree: Authentication Path
6.3 Dynamic Tree: Authentication Path
Chapter 7: Entropy of Peer-to-Peer Network
1 Blockchain Network, Blockchain Transaction Network, and Peer-to-Peer Network
2 The Shannon Entropy and the Entropy Rate for the Peer-to-Peer Network
2.1 Shannon Entropy and Entropy Rate
2.2 Markov Traceability
2.3 Maximal Entropy Random Walk
3 Privacy and the Blockchain Transaction Graph
3.1 The Privacy Issue
3.2 A Privacy Violation Model
4 The Particles Model and the Entropy of Privacy
4.1 Entropy of Privacy
4.2 The Particles Model
4.3 Average Entropy
4.4 Privacy Entropy Rate
Chapter 8: Applications: Selection of Research Studies
1 Satoshi Nakamoto´s White Paper: On the Probability of a 51% Attack
2 On forecasting Bitcoin Price with Chainlets
3 On Valuing Bitcoin Price with Metcalfe´s Law
4 On Valuing Bitcoin Price from an Equilibrium Approach
5 Weights Contribution of a Crypto Portfolio through the Euler Allocation
6 On Tracing Knowledge Publicly: The MathCoin as an Example
Conclusion
References
Index
