Some Fundamentals of Mathematics of Blockchain

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