Intro
Preface
Contents
Chapter 1 Introduction
Chapter 2 Mathematical Background
2.1 Number Theory
2.1.1 Divisibility and Factorization
2.1.2 Greatest Common Divisor (GCD)
2.1.3 Modular Arithmetic
2.1.4 Modular Exponentiation
2.1.5 Modular Inverse
2.1.6 Euler's Theorem
2.1.7 Carmichael's Theorem
2.1.8 Generators
2.1.9 Chinese Remainder Theorem
2.1.10 Quadratic Residues
2.1.11 Higher-Order Residues
2.1.12 Residue Classes
2.1.13 Random Number Generators
2.2 Group Theory
2.2.1 Basic Axioms
2.2.2 Multiplicative Groups

2.2.3 Additive Groups
2.2.4 Order of a Group
2.2.5 Cyclic Groups and Subgroups
2.2.6 Discrete Logarithm Problem
2.2.7 Diffie-Hellman Problem
2.2.8 Cosets and Lagrange's Theorem
2.2.9 Sylow's Theorem
2.2.10 Direct Products
2.2.11 Homomorphisms
2.3 Field Theory
2.4 Elliptic Curves
2.4.1 Group Law
2.4.2 Elliptic Curve Point Multiplication
2.4.3 Elliptic Curves over Finite Fields
Chapter 3 Rivest-Shamir-Adleman Algorithm
3.1 Key Generation
3.2 Encryption
3.3 Decryption
3.4 Homomorphic Properties
3.5 Security
3.6 Example

Chapter 4 Goldwasser-Micali Algorithm
4.1 Key Generation
4.2 Encryption
4.3 Decryption
4.4 Homomorphic Properties
4.5 Security
4.6 Example
Chapter 5 ElGamal Algorithm
5.1 Multiplicatively Homomorphic ElGamal Algorithm
5.1.1 Key Generation
5.1.2 Encryption
5.1.3 Decryption
5.1.14 Homomorphic Properties
5.1.5 Security
5.1.6 Example
5.2 Additively Homomorphic ElGamal
5.2.1 Encryption
5.2.2 Decryption
5.2.3 Homomorphic Properties
5.3 The Elliptic Curve ElGamal Algorithm
5.3.1 Key Generation
5.3.2 Encryption
5.3.3 Decryption

5.3.4 Homomorphic Properties
Chapter 6 Benaloh Algorithm
6.1 Key Generation
6.2 Encryption
6.3 Decryption
6.4 Homomorphic Properties
6.5 Security
6.6 Example
Chapter 7 Naccache-Stern Algorithm
7.1 The Deterministic Version
7.1.1 Key Generation
7.1.2 Encryption
7.1.3 Decryption
7.1.4 Security
7.2 The Probabilistic Version
7.2.1 Key Generation
7.2.2 Encryption
7.2.3 Decryption
7.2.4 Homomorphic Properties
7.2.5 Security
7.2.6 Example
Chapter 8 Okamoto-Uchiyama Algorithm
8.1 Key Generation
8.2 Encryption
8.3 Decryption

8.4 Homomorphic Properties
8.5 Security
8.6 Example
Chapter 9 Paillier Algorithm
9.1 Key Generation
9.2 Encryption
9.3 Decryption
9.4 Homomorphic Properties
9.5 Security
9.6 Example
Chapter 10 Damgård-Jurik Algorithm
10.1 Forming the Plaintext Space
10.2 Key Generation
10.3 Encryption
10.4 Decryption
10.5 Homomorphic Properties
10.6 Security
10.7 Example
Chapter 11 Boneh-Goh-Nissim Algorithm
11.1 Key Generation
11.2 Encryption
11.3 Decryption
11.4 Homomorphic Properties
11.5 Security
11.6 Example