
Introduction to Identity-Based Encryption
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Content
- Intro
- Contents
- Preface
- 1 Introduction
- 1.1 What Is IBE?
- 1.2 Why Should I Care About IBE?
- 2 Basic Mathematical Concepts and Properties
- 2.1 Concepts from Number Theory
- 2.1.1 Computing the GCD
- 2.1.2 Computing Jacobi Symbols
- 2.2 Concepts from Abstract Algebra
- 3 Properties of Elliptic Curves
- 3.1 Elliptic Curves
- 3.2 Adding Points on Elliptic Curves
- 3.2.1 Algorithm for Elliptic Curve Point Addition
- 3.2.2 Projective Coordinates
- 3.2.3 Adding Points in Jacobian Projective Coordinates
- 3.2.4 Doubling a Point in Jacobian Projective Coordinates
- 3.3 Algebraic Structure of Elliptic Curves
- 3.3.1 Higher Degree Twists
- 3.3.2 Complex Multiplication
- 4 Divisors and the Tate Pairing
- 4.1 Divisors
- 4.1.1 An Intuitive Introduction to Divisors
- 4.2 The Tate Pairing
- 4.2.1 Properties of the Tate Pairing
- 4.3 Miller's Algorithm
- 5 Cryptography and Computational Complexity
- 5.1 Cryptography
- 5.1.1 Definitions
- 5.1.2 Protection Provided by Encryption
- 5.1.3 The Fujisaki-Okamoto Transform
- 5.2 Running Times of Useful Algorithms
- 5.2.1 Finding Collisions for a Hash Function
- 5.2.2 Pollard's Rho Algorithm
- 5.2.3 The General Number Field Sieve
- 5.2.4 The Index Calculus Algorithm
- 5.2.5 Relative Strength of Algorithms
- 5.3 Useful Computational Problems
- 5.3.1 The Computational Diffie-Hellman Problem
- 5.3.2 The Decision Diffie-Hellman Problem
- 5.3.3 The Bilinear Diffie-Hellman Problem
- 5.3.4 The Decision Bilinear Diffie-Hellman Problem
- 5.3.5 q-Bilinear Diffie-Hellman Inversion
- 5.3.6 q-Decision Bilinear Diffie-Hellman Inversion
- 5.3.7 Cobilinear Diffie-Hellman Problems
- 5.3.8 Integer Factorization
- 5.3.9 Quadratic Residuosity
- 5.4 Selecting Parameter Sizes
- 5.4.1 Security Based on Integer Factorization and Quadratic Residuosity
- 5.4.2 Security Based on Discrete Logarithms
- 5.5 Important Special Cases
- 5.5.1 Anomalous Curves
- 5.5.2 Supersingular Elliptic Curves
- 5.5.3 Singular Elliptic Curves
- 5.5.4 Weak Primes
- 5.6 Proving Security of Public-Key Algorithms
- 5.7 Quantum Computing
- 5.7.1 Grover's Algorithm
- 5.7.2 Shor's Algorithm
- 6 Related Cryptographic Algorithms
- 6.1 Goldwasser-Michali Encryption
- 6.2 The Diffie-Hellman Key Exchange
- 6.3 Elliptic Curve Diffie-Hellman
- 6.4 Joux's Three-Way Key Exchange
- 6.5 ElGamal Encryption
- 7 The Cocks IBE Scheme
- 7.1 Setup of Parameters
- 7.2 Extraction of the Private Key
- 7.3 Encrypting with Cocks IBE
- 7.4 Decrypting with Cocks IBE
- 7.5 Examples
- 7.6 Security of the Cocks IBE Scheme
- 7.6.1 Relationship to the Quadratic Residuosity Problem
- 7.6.2 Chosen Ciphertext Security
- 7.6.3 Proof of Security
- 7.6.4 Selecting Parameter Sizes
- 7.7 Summary
- 8 Boneh-Franklin IBE
- 8.1 Boneh-Franklin IBE (Basic Scheme)
- 8.1.1 Setup of Parameters (Basic Scheme)
- 8.1.2 Extraction of the Private Key (Basic Scheme)
- 8.1.3 Encrypting with Boneh-Franklin IBE (Basic Scheme)
- 8.1.4 Decrypting with Boneh-Franklin IBE (Basic Scheme)
- 8.1.5 Examples (Basic Scheme)
- 8.2 Boneh-Franklin IBE (Full Scheme)
- 8.2.1 Setup of Parameters (Full Scheme)
- 8.2.2 Extraction of the Private Key (Full Scheme)
- 8.2.3 Encrypting with Boneh-Franklin IBE (Full Scheme)
- 8.2.4 Decrypting with Boneh-Franklin IBE (Full Scheme)
- 8.3 Security of the Boneh-Franklin IBE Scheme
- 8.4 Summary
- 9 Boneh-Boyen IBE
- 9.1 Boneh-Boyen IBE (Basic Scheme-Additive Notation)
- 9.1.1 Setup of Parameters (Basic Scheme-Additive Notation)
- 9.1.2 Extraction of the Private Key (Basic Scheme-Additive Notation)
- 9.1.3 Encrypting with Boneh-Boyen IBE (Basic Scheme-Additive Notation)
- 9.1.4 Decrypting with Boneh-Boyen IBE (Basic Scheme-Additive Notation)
- 9.2 Boneh-Boyen IBE (Basic Scheme-Multiplicative Notation)
- 9.2.1 Setup of Parameters (Basic Scheme-Multiplicative Notation)
- 9.2.2 Extraction of the Private Key (Basic Scheme-Multiplicative Notation)
- 9.2.3 Encrypting with Boneh-Boyen IBE (Basic Scheme-Multiplicative Notation)
- 9.2.4 Decrypting with Boneh-Boyen IBE (Basic Scheme- Multiplicative Notation)
- 9.3 Boneh-Boyen IBE (Full Scheme)
- 9.3.1 Setup of Parameters (Full Scheme)
- 9.3.2 Extraction of the Private Key (Full Scheme)
- 9.3.3 Encrypting with Boneh-Boyen IBE (Full Scheme)
- 9.3.4 Decrypting with Boneh-Boyen IBE (Full Scheme)
- 9.4 Security of the Boneh-Boyen IBE Scheme
- 9.5 Summary
- 10 Sakai-Kasahara IBE
- 10.1 Sakai-Kasahara IBE (Basic Scheme-Additive Notation)
- 10.1.1 Setup of Parameters (Basic Scheme-Additive Notation)
- 10.1.2 Extraction of the Private Key (Basic Scheme-Additive Notation)
- 10.1.3 Encrypting with Sakai-Kasahara IBE (Basic Scheme-Additive Notation)
- 10.1.4 Decrypting with Sakai-Kasahara IBE (Basic Scheme-Additive Notation)
- 10.2 Sakai-Kasahara IBE (Basic Scheme-Multiplicative Notation)
- 10.2.1 Setup of Parameters (Basic Scheme-Multiplicative Notation)
- 10.2.2 Extraction of the Private Key (Basic Scheme-Multiplicative Notation
- 10.2.3 Encrypting with Sakai-Kasahara IBE (Basic Scheme- Multiplicative Notation)
- 10.2.4 Decrypting with Sakai-Kasahara IBE (Basic Scheme- Multiplicative Notation)
- 10.3 Sakai-Kasahara IBE (Full Scheme)
- 10.3.1 Setup of Parameters (Full Scheme)
- 10.3.2 Extraction of the Private Key (Full Scheme)
- 10.3.3 Encrypting with Sakai-Kasahara IBE (Full Scheme)
- 10.3.4 Decrypting with Sakai-Kasahara IBE (Full Scheme)
- 10.4 Security of the Sakai-Kasahara IBE Scheme
- 10.5 Summary
- 11 Hierarchical IBE and Master Secret Sharing
- 11.1 HIBE Based on Boneh-Franklin IBE
- 11.1.1 GS HIBE (Basic) Root Setup
- 11.1.2 GS HIBE (Basic) Lower-Level Setup
- 11.1.3 GS HIBE (Basic) Extract
- 11.1.4 GS HIBE (Basic) Encrypt
- 11.1.5 GS HIBE (Basic) Decrypt
- 11.2 Example of a GS HIBE System
- 11.2.1 GS HIBE (Basic) Root Setup
- 11.2.2 GS HIBE (Basic) Lower-Level Setup
- 11.2.3 GS HIBE (Basic) Extraction of Private Key
- 11.2.4 GS HIBE (Basic) Encryption
- 11.2.5 GS HIBE (Basic) Decryption
- 11.3 HIBE Based on Boneh-Boyen IBE
- 11.3.1 BBG HIBE (Basic) Setup
- 11.3.2 BBG HIBE (Basic) Extract
- 11.3.3 BBG HIBE (Basic) Encryption
- 11.3.4 BBG HIBE (Basic) Decryption
- 11.4 Example of a BBG HIBE System
- 11.4.1 BBG HIBE (Basic) Setup
- 11.4.2 BBG HIBE (Basic) Extraction of Private Key
- 11.4.3 BBG HIBE (Basic) Encryption
- 11.4.4 BBG HIBE (Basic) Decryption
- 11.5 Master Secret Sharing
- 11.6 Master Secret Sharing Example
- 12 Calculating Pairings
- 12.1 Pairing-Friendly Curves
- 12.1.1 Relative Efficiency of Parameters of Pairing-Friendly Curves
- 12.2 Eliminating Irrelevant Factors
- 12.2.1 Eliminating Random Components
- 12.2.2 Eliminating Extension Field Divisions
- 12.2.3 Denominator Elimination
- 12.3 Calculating the Product of Pairings
- 12.4 The Shipsey-Stange Algorithm
- 12.5 Precomputation
- Appendix: Useful Test Data
- About the Author
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