Functional Encryption

Foreword
Preface
Contents
1 Foundations of Functional Encryption
	1 Introduction
	2 Functional Encryption Syntax
		2.1 Functionality
		2.2 Functional Encryption Scheme
	3 Relationship Between Functional Encryption, Obfuscation, Fully Homomorphic Encryption
	4 Challenges
	5 Applications
	6 Present and Future Directions in Functional Encryption
	7 Additive Inverse
		7.1 Modular Arithmetic
		7.2 Addition and Subtraction Property of Modular Arithmetic
	8 Multiplicative Inverse
		8.1 Multiplication Property of Modular Arithmetic
	9 Matrices
	10 Linear Congruence
	11 Prime and Relative Prime Numbers
	12 Greatest Common Divisor (Euclid\'s Algorithm, Bezout\'s Algorithm, Extended Euclid\'s Algorithm)
		12.1 Greatest Common Divisor (GCD)
		12.2 Euclidean/ Euclid\'s Algorithm
		12.3 Bezout\'s Algorithm
		12.4 Extended Euclid\'s Algorithm
	13 Conclusion and Future Work
	References
2 Impact of Group Theory in Cryptosystem
	1 Introduction
	2 Group Theory
		2.1 Group Axiom
		2.2 Illustrations
		2.3 Properties of Group
			2.3.1 Abelian Group
			2.3.2 Lagrange\'s Theorem
			2.3.3 Schnorr Group
			2.3.4 Finite Field
		2.4 Properties of Finite Field
		2.5 Bilinear Mapping
	3 Mathematical Tools
		3.1 Functionalities (Equality Test, Predicate Encryption, Inequality Test, Inner Product Evaluation) and Other Methods
			3.1.1 Equality Test
			3.1.2 Inequality Test
			3.1.3 Inner Product Evaluation
		3.2 Wiener Theorem
		3.3 Primality Test \"026A30C  Set 1 (Introduction and School Method)
			3.3.1 School Method
			3.3.2 Fermat\'s Strategy
			3.3.3 Fermat\'s Little Theorem
			3.3.4 Miller-Rabin Method
			3.3.5 The Solovay–Strassen Primality Test
		3.4 Discrete Logarithm Problem in Subgroup of  Zp
		3.5 Baby-Step Giant-Step Algorithm
		3.6 Functional Field Sieve
		3.7 Elliptic Curve Factorization
		3.8 Random Square Factoring
		3.9 Quadratic Sieve Method
		3.10 Factoring Polynomials over a Finite Field
		3.11 Irreducible Polynomials over a Finite Field Zp
	4 Role of Group Theory in Cryptosystem
		4.1 Use of Group Theory in RSA-Based Cryptosystem
		4.2 Schnorr Group in Log-Based Cryptosystem
		4.3 Elliptic Curve and Bilinear Mapping-Based Cryptosystem
	5 Conclusions
	References
3 XTR Algorithm: Efficient and Compact Subgroup Trace Representation
	1 Introduction
	2 Some Definitions and Results from Finite Field
	3 Fundamental of XTR
		3.1 XTR Group
		3.2 Arithmetic Operation in GF(p2)
		3.3 Translation of Arithmetic Operation of G to GF(p2)
		3.4 Parameter Selection
		3.5 Subgroup Selection
	4 Cryptographic Applications of XTR
		4.1 XTR Version of DH Problem and its Variants
		4.2 XTR-Diffie-Hellman (XTR-DH) Key Exchange Scheme
		4.3 XTR-ElGamal Encryption Scheme
		4.4 XTR-Nyberg-Rueppel (NR) Signature Scheme
		4.5 XTR-DSA Signature Scheme
	5 Recent Development of XTR
	6 Conclusion
	References
4 HECC (Hyperelliptic Curve Cryptography)
	1 Introduction
		1.1 Basic Mathematical Terminologies
			1.1.1 Arithmetical Closure
			1.1.2 The Cryptography of Hyperelliptic Curve
			1.1.3 Finite Field
			1.1.4 Interpretation
		1.2 Divisors
			1.2.1 Explanation 1 (Divisor, Degree, Order at a Point)
			1.2.2 Explanation 2 (GCD of Divisors)
			1.2.3 Explanation 3 (Rational Function Divisor)
			1.2.4 Explanation 4 (Primary Divisor)
			1.2.5 Explanation 5 (Divisor of Jacobian)
		1.3 The Jacobian of Hyperelliptic Curve
			1.3.1 A Jacobian Instruction
			1.3.2 Theory of Hasse-Weil
			1.3.3 Specification of Zeta Function
			1.3.4 Methodology for Ordering Genus 2 of the Jacobian Hyperelliptic Curve
		1.4 Semi-Reduced and Reduced Divisors
			1.4.1 The Accession of Reduced Divisors
			1.4.2 Methodology for Extension Euclidean Polynomial Methods
			1.4.3 Methodology for Accession
		1.5 Semi-Reduced Sums Via Mumford Arithmetic
		1.6 Reduction Via Mumford Arithmetic
	2 Hyperelliptic Curves over Finite Fields
		2.1 Proof of Hyperelliptic Curve
	3 Hyperelliptic Curve Key Pair Generation
		3.1 Divisor Order
		3.2 Principle of Hyperelliptic Curve Cryptographic Arrangement
	4 Hyperelliptic Curve Encryption and Decryption
		4.1 Procedure 1. Hyperelliptic Curve Encryption
		4.2 Procedure 2. Hyperelliptic Curve Decryption
	5 HECC Signature Algorithm
		5.1 Elgamal Signature Method
		5.2 HECC Signature Generation
		5.3 HECC Signature Verification
	6 Security of HECC
	7 Conclusion
	References
5 Pairing-Based Cryptography
	1 Introduction
	2 Mathematical Terms and Concepts Used
		2.1 Random Oracle
		2.2 Symmetric Encryption
		2.3 Asymmetric Encryption
		2.4 Public Key Encryption
		2.5 Subexponential Algorithm
		2.6 ECDLP (EC Discrete Logarithm Problem)
		2.7 Key Escrow
		2.8 Turing Machine
		2.9 Elliptic Curves
		2.10 Jacobian of Hyper ECs
		2.11 Group [2]
		2.12 Field [3, 4]
		2.13 Finite Field
		2.14 Diffie–Hellman Key Exchange and the Problem
		2.15 Miller\'s Algorithm
		2.16 Elliptic Curve Cryptography (ECC) over R
	3 Pairings and Different Pairing Methods
		3.1 Weil Pairing
		3.2 Tate-Lichtenbaum Pairing
		3.3 Hyperelliptic Tate-Lichtenbaum Pairing [12, 17]
		3.4 Eta Pairing
		3.5 Ate Pairing
	4 Drawbacks/Vulnerabilities of PBC
	5 Security of Pairing-Based Cryptography
		5.1 Fault Attacks
			5.1.1 Fault against Duursma–Lee
			5.1.2 Attacks against the Miller\'s Algorithm
		5.2 Side Channel Analysis
			5.2.1 Timing Attacks
			5.2.2 Power Analysis
	6 Functional Encryption and its Impact on Cryptography
	7 Conclusion and Future Advancements
	References
6 NTRU Algorithm: Nth Degree Truncated Polynomial Ring Units
	1 Introduction
	2 NTRU Cryptosystem
		2.1 Symbols and Notations
		2.2 Parameters
		2.3 Algorithm
		2.4 Key Generation
		2.5 Encryption
		2.6 Decryption
		2.7 Working Method
	3 NTRU Optimization
	4 Security
	5 Improvements
	6 Conclusion
	References
7 Cocks IBE Scheme
	1 Introduction
		1.1 Identity-Based Encryption (IBE)
		1.2 Cocks IBE
		1.3 Working of Cocks IBE
		1.4 Features of Cocks IBE
	2 Mathematical Concepts
	3 Setting up of Parameters
		3.1 How it Works
	4 Extraction of Private Key
		4.1 How it Works
	5 Encryption
		5.1 How it Works
	6 Decryption
		6.1 How it Works
	7 Examples
	8 Correctness of Cocks IBE
	9 Security
		9.1 Quadratic Residuosity Problem
		9.2 Chosen Ciphertext Security
		9.3 Proof of Security
	10 Summary
	References
8 Boneh-Franklin IBE
	1 Introduction
		1.1 Identity-Based Encryption (IBE)
	2 Boneh-Franklin (Basic Scheme)
		2.1 Setting up the Parameters
		2.2 (Basic Scheme) Private Key Extraction
		2.3 Encryption (Basic Scheme)
		2.4 Decryption (Basic Scheme)
	3 Boneh-Franklin IBE (Full Scheme)
		3.1 Setup of Parameters
		3.2 (Full Scheme) Private Key Extraction
		3.3 Encryption (Full Scheme)
		3.4 Decryption (Full Scheme)
	4 Security
	5 Further Works on Boneh-Franklin IBE
	6 Examples
	7 Conclusion
	References
9 Boneh-Boyen IBE
	1 Introduction
		1.1 Limitation
		1.2 Identity-Based Encryption (IBE)
		1.3 Setup Algorithm
		1.4 Extract Algorithm
		1.5 Encryption Algorithm
		1.6 Decryption Algorithm
		1.7 Boneh-Boyen IBE
		1.8 Classification of IBE Schemes
			1.8.1 “Quadratic Residuosity” IBE
			1.8.2 “Full Domain Hash” IBE
			1.8.3 “Exponent Inversion” IBE
			1.8.4 “Commutative Blinding” IBE
	2 Boneh-Boyen IBE (Basic Scheme: Additive Notation)
		2.1 Additive Notation
		2.2 Setup of Parameters (Basic Scheme: Additive Notation)
		2.3 Extraction of the Private Key (Basic Scheme: Additive Notation)
		2.4 Encryption with Boneh-Boyen IBE (Basic Scheme: Additive Notation)
		2.5 Decrypting with Boneh-Boyen IBE (Basic Scheme: Additive Notation)
	3 Boneh-Boyen IBE (Basic Scheme: Multiplicative Notation)
		3.1 Setup of Parameters (Basic Scheme: Multiplicative Notation)
		3.2 Extraction of the Private Key (Basic Scheme: Multiplicative Notation)
		3.3 Encrypting with Boneh-Boyen IBE (Basic Scheme: Multiplicative Notation)
		3.4 Decrypting with Boneh-Boyen IBE (Basic Scheme: Multiplicative Notation)
	4 Boneh-Boyen IBE (Full Scheme)
		4.1 Setup of Parameters (Full Scheme)
		4.2 Extraction of the Private Key (Full Scheme)
		4.3 Encrypting with Boneh-Boyen IBE (Full Scheme)
		4.4 Decrypting with Boneh-Boyen IBE (Full Scheme)
	5 Security of the Boneh-Boyen IBE Scheme
	6 Conclusion and Future Scope
	References
10 Sakai-Kasahara IBE
	1 Introduction
	2 Sakai-Kasahara IBE (Basic Scheme: Additive Notation)
		2.1 Setup of Parameters
		2.2 Extraction of the Private Key
		2.3 Sakai-Kasahara IBE Encryption
		2.4 Sakai-Kasahara IBE Decryption
	3 Sakai-Kasahara IBE (Basic Scheme: Multiplicative Notation)
		3.1 Setup of Parameters
		3.2 Extraction of the Private Key
		3.3 Sakai-Kasahara IBE Encryption
		3.4 Sakai-Kasahara IBE Decryption
	4 Sakai-Kasahara IBE (Full Scheme)
		4.1 Setup of Parameters
		4.2 Extraction of the Private Key
		4.3 Sakai-Kasahara IBE Encryption
		4.4 Sakai-Kasahara IBE Decryption
	5 Security of the Sakai-Kasahara IBE Scheme
	References
11 HIBE: Hierarchical Identity-Based Encryption
	1 Introduction
		1.1 Public Key Cryptography
		1.2 Pitfalls of Public Key Cryptosystems
		1.3 Identity-Based Encryption
		1.4 Hierarchical Identity-Based Encryption
	2 HIBE Based on Boneh-Franklin IBE
		2.1 HIBE (Based on Boneh-Franklin IBE) Root Setup
		2.2 HIBE (Based on Boneh-Franklin IBE) Lower Level Setup
		2.3 HIBE (Based on Boneh-Franklin IBE) Extraction of the Private Key
		2.4 HIBE (Based on Boneh-Franklin IBE) Encryption
		2.5 HIBE (Based on Boneh-Franklin IBE) Decryption
	3 HIBE Based on Boneh-Boyen IBE
		3.1 HIBE (Based on Boneh-Boyen IBE) Setup
		3.2 HIBE (Based on Boneh-Boyen IBE) Extraction of the Private Key
		3.3 HIBE (Based on Boneh-Boyen IBE) Encryption
		3.4 HIBE (Based on Boneh-Boyen IBE) Decryption
	4 Master Secret Sharing
	5 Security of the Hierarchical IBE
	References
12 Extensions of IBE and Related Primitives
	1 Introduction
	2 Identity-Based Key Agreement (IBKA)
		2.1 Formal Model
		2.2 Security Requirements
			2.2.1 Oracles
	3 Fuzzy Identity-Based Encryption (FIBE)
		3.1 Formal Model
		3.2 Security Properties
	4 Threshold Broadcast Encryption with Keyword Search
		4.1 Formal Model
		4.2 Security Requirements
	5 Linear Encryption with Keyword Search
		5.1 Formal Model
		5.2 Security Requirements
	6 Public Key Encryption with Keyword Search
		6.1 Formal Model
		6.2 Security Requirements
	7 Identity-Based Encryption with Wildcards (WIBE)
		7.1 Formal Model
		7.2 Security Requirements
	8 Identity-Based Encryption with Wildcard Key Derivation (WKD-IBE)
		8.1 Formal Model
		8.2 Security Requirements
	9 Identity-Based Conditional Proxy Re-Encryption (IBCPRE)
		9.1 Formal Model
		9.2 Security Requirements
	10 Summary
	References
13 Attribute-Based Encryption
	1 Introduction
		1.1 Encryption
		1.2 Public Key Encryption
			1.2.1 Limitation
		1.3 Public Key Infrastructure (PKI)
		1.4 Identity-Based Encryption (IBE)
	2 Attribute-Based Encryption (ABE)
		2.1 Basic Model of ABE
			2.1.1 Complexity Assumptions
		2.2 Access Policy
		2.3 ABE Working
	3 ABE Categories
		3.1 Key Policy-ABE (KP-ABE)
		3.2 Ciphertext Policy-ABE (CP-ABE)
	4 Difference between CP-ABE and KP-ABE
	5 Literature Review of KP-ABE and CP-ABE
	6 Comparison of ABE, KP-ABE, CP-ABE
	7 The Security Model
		7.1 Fuzzy Model for ABE
	8 Conclusion
	References
14 Digital Signatures
	1 Introduction
		1.1 Security Models for Signature Schemes
		1.2 Signature Schemes with Hash Functions
		1.3 Organization of the Chapter
	2 Signature Schemes Based on IFP and DLP
		2.1 RSA Signature Scheme
		2.2 ElGamal Signature Scheme
		2.3 The Schnorr Signature Algorithm
		2.4 Nuberg-Rueppel Digital Signature Algorithm
		2.5 Digital Signature Standard (DSS)
	3 Blind Signature Scheme
		3.1 Chaum\'s RSA Blind Signature Scheme
		3.2 Schnorr Blind Signature Protocol
		3.3 Okomoto-Schnorr Blind Signature Protocol
	4 Undeniable Signature Schemes
		4.1 Chaum-Van Antwerper Undeniable Signature Scheme
		4.2 RSA-Based Undeniable Signature Schemes
	5 The Short Signature Scheme
		5.1 The Boneh-Lynn-Shacham Short Signature Scheme
			5.1.1 Bilinear Pairing [51]
			5.1.2 Computation Assumptions [51]
			5.1.3 The Boneh-Lynn-Shacham Short Signature Scheme
		5.2 The Boneh-Boyen Short Signature Scheme
			5.2.1 Computational Assumption
			5.2.2 The Boneh-Boyen Short Signature Scheme
	6 Hierarchical Identity-Based Signature Scheme
	7 Other Signature Schemes
	8 Signcryption
	9 Conclusion
	References
15 QUIET: Quatro-Inverse Exponential Cipher Technique
	1 Introduction
	2 Problem Statement
	3 Proposed Cryptosystem
	4 Methodology Adopted
	5 Key Selection
	6 Key Sharing
	7 Encoding and Decoding Operation
		7.1 Encryption Operation
		7.2 Cipher Text Decoding: Decryption Operation
	8 Algorithm Formulated
		8.1 Key Generation Algorithm
		8.2 Encryption Algorithm
	9 Encryption Simulation
	10 Encryption Flowchart
	11 Result
	12 Conclusion and Future Work
	References
Index