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Cryptographic Primitives in Blockchain Technology

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Introduction

Foreword About the Course

Preliminaries

Euler’s Totient Function 𝜑(n)

Modular Arithmetic

Groups: Elementary Properties

The Group of Units

Computational Complexity

Asymptotic Order of Growth

Classes of Time Complexity

Efficiency of Algorithms

Quiz: Preliminaries

Cryptographic Primitives

Introduction Cryptography Basics A Basic Cryptosystem Symmetric vs. Asymmetric Cryptography Public-Key Cryptography The Basic Functionality of a Public-Key Cryptosystem The Diffie-Hellman Key Exchange The Discrete Logarithm Problem Hash Functions The Security of Hash Functions and the Birthday Attack Real Hash Functions Real Hash Functions: Structural Weakness Merkle Trees Quiz: Cryptographic Primitives

Elliptic Curve Cryptography

Introduction Weierstrass Equations Elliptic Curves Group Law Elliptic Curves Over Finite Fields Addition of Points Group Structure The Elliptic Curve Discrete Logarithm Problem Attacking the ECDLP Cryptographically Secure Elliptic Curves Additional Security Requirements Recommended Elliptic Curves by Standards The Elliptic Curve secp256k1 The Elliptic Curve Digital Signature Algorithm Attacks on ECDSA Quiz: Elliptic Curves Cryptography

Information Security in Software Systems

The CIA Triad Attacks Against Distributed Systems Proof-of-Work Quiz: Information Security in Software Systems

Distributed Systems

Introduction Classification of Systems Centralized/Decentralized Systems Distributed Systems CAP theorem Peer-to-Peer Systems Reaching Consensus in Distributed Systems Models of Computation Consensus and Agreement Protocols Basic Properties of Consensus Protocols Agreement in a Failure-Free System Consensus with Crash Failures in Synchronous Systems The Byzantine Agreement in Synchronous Systems Problem Definition Agreement in Asynchronous Systems Practical Solutions Summary Quiz: Distributed Systems

Introduction to Blockchain Technology

Introduction Definition of Blockchain Blockchain as a Suit of Technology Ordering Transactions in a Blockchain Implementing State Replication by Using Blockchain Reaching Consensus on the Network Resolving Forks Using the Longest-Chain-Criterion Acceptance of New Blocks The Blockchain as a Data Structure The Blockchain Information Security in Blockchain Systems Proof-of-Work The Double-Spending Attack The General Double-Spending Attack: The Race Attack Other Attacks The Incentive System of the Nakamoto Consensus Summary Quiz: Introduction to Blockchain Technology

Bitcoin

Introduction System basics, Keys, and Addresses Transactions The Transaction Script Lifecycle of Transactions The Blockchain Summary Quiz: Bitcoin

Introduction to Quantum Computing

Introduction Quantum Computation Algorithms Impact on Present Cryptography Summary Quiz: Introduction to Quantum Computing

Bitcoin Under Broken Crypto Primitives

A General Overview of the Primitives Hash Functions Impact of Quantum Computers Summary Quiz: Bitcoin Under Broken Crypto Primitives

Post-Quantum Blockchains

Introduction Post-Quantum Cryptography Comparison Between Classical and Quantum Safe Schemes Hash-Based Cryptosystems Summary Quiz: Post-Quantum Blockchain

In Closing

Conclusion Copyright Information

Finite Fields

Learn about finite fields and their properties, prime fields, and primitive roots in this lesson.

We'll cover the following

Introduction
Existence and uniqueness
Basic properties
Prime fields
- Lemma 2: proper subgroups
The multiplicative group and primitive roots

Introduction

Finite fields or Galois fields, denoted by $\mathbb{F}$ or $G F$ , are fields that only contain a finite number of elements. Any finite field $\mathbb{F}$ is equipped with two operations, namely addition and multiplication, which satisfy the properties of the definition :Field_Definition. The number of elements of $\mathbb{F}$ is called the order of $\mathbb{F}$ .

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