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Introduction 11.1 A Very Brief and
Introduction 1
1.1 A Very Brief and Informal Overview of Our Construction . . . . . . . . . . 2
1.2 What is Fully Homomorphic Encryption? . . . . . . . . . . . . . . . . . . . 5
1.3 Bootstrapping a Scheme that Can Evaluate its Own Decryption Circuit . . 7
1.4 Ideal Lattices: Ideally Suited to Construct Bootstrappable Encryption . . . 10
1.5 Squashing the Decryption Circuit: The Encrypter Starts Decryption! . . . . 15
1.6 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Definitions related to Homomorphic Encryption 27
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Computational Security Definitions . . . . . . . . . . . . . . . . . . . . . . . 31
3 Previous Homomorphic Encryption Schemes 34
4 Bootstrappable Encryption 43
4.1 Leveled Fully Homomorphic Encryption from Bootstrappable Encryption, Generically 43
4.2 Correctness, Computational Complexity and Security of the Generic Construction 48
4.3 Fully Homomorphic Encryption from KDM-Secure Bootstrappable Encryption 51
4.4 Fully Homomorphic Encryption from Bootstrappable Encryption in the Random Oracle Model 53
vi
5 An Abstract Scheme Based on the Ideal Coset Problem 57
5.1 The Ideal Coset Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 An Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Security of the Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Background on Ideal Lattices I: The Basics 63
6.1 Basic Background on Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Basic Background on Ideal Lattices . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Probability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 A Somewhat Homomorphic Encryption Scheme 69
7.1 Why Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Why Ideal Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 A Geometric Approach to Maximizing the Circuit Depth that Can Be Evaluated 70
7.4 Instantiating the Ring: The Geometry of Polynomial Rings . . . . . . . . . 72
7.5 Instantiating Encrypt and Minimizing rEnc . . . . . . . . . . . . . . . . . . . 75
7.6 Instantiating Decrypt and Maximizing rDec . . . . . . . . . . . . . . . . . . . 75
7.7 Security of the Concrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . 77
7.8 How Useful is the Somewhat Homomorphic Scheme By Itself? . . . . . . . . 79
8 Tweaks to the Somewhat Homomorphic Scheme 81
8.1 On the Relationship between the Dual and the Inverse of an Ideal Lattice . 82
8.2 Transference Lemmas for Ideal Lattices . . . . . . . . . . . . . . . . . . . . 85
8.3 Tweaking the Decryption Equation . . . . . . . . . . . . . . . . . . . . . . . 86
8.4 A Tweak to Reduce the Circuit Complexity of the Rounding Step in Decryption 88
9 Decryption Complexity of the Tweaked Scheme 90
10 Squashing the Decryption Circuit 98
10.1 A Generic Description of the Transformation . . . . . . . . . . . . . . . . . 98
10.2 How to Squash, Concretely . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.3 Bootstrapping Achieved: The Decryption Circuit for the Transformed System 102
11 Security 104
11.1 Regarding the Hint Given in Our “Squashing” Transformation . . . . . . . 104
vii
11.2 Counterbalancing Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 113
12 Performance and Optimizations 115
12.1 Simple Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.2 Basic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.3 More Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13 Background on Ideal Lattices II 125
13.1 Overview of Gaussian Distributions over Lattices . . . . . . . . . . . . . . . 125
13.2 The Smoothing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
13.3 Sampling a Lattice According to a Gaussian Distribution . . . . . . . . . . 128
13.4 Ideal Factorization in Polynomial Rings . . . . . . . . . . . . . . . . . . . . 129
1.1 A Very Brief and Informal Overview of Our Construction . . . . . . . . . . 2
1.2 What is Fully Homomorphic Encryption? . . . . . . . . . . . . . . . . . . . 5
1.3 Bootstrapping a Scheme that Can Evaluate its Own Decryption Circuit . . 7
1.4 Ideal Lattices: Ideally Suited to Construct Bootstrappable Encryption . . . 10
1.5 Squashing the Decryption Circuit: The Encrypter Starts Decryption! . . . . 15
1.6 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Definitions related to Homomorphic Encryption 27
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Computational Security Definitions . . . . . . . . . . . . . . . . . . . . . . . 31
3 Previous Homomorphic Encryption Schemes 34
4 Bootstrappable Encryption 43
4.1 Leveled Fully Homomorphic Encryption from Bootstrappable Encryption, Generically 43
4.2 Correctness, Computational Complexity and Security of the Generic Construction 48
4.3 Fully Homomorphic Encryption from KDM-Secure Bootstrappable Encryption 51
4.4 Fully Homomorphic Encryption from Bootstrappable Encryption in the Random Oracle Model 53
vi
5 An Abstract Scheme Based on the Ideal Coset Problem 57
5.1 The Ideal Coset Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 An Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Security of the Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Background on Ideal Lattices I: The Basics 63
6.1 Basic Background on Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Basic Background on Ideal Lattices . . . . . . . . . . . . . . . . . . . . . . . 65
6.3 Probability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7 A Somewhat Homomorphic Encryption Scheme 69
7.1 Why Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Why Ideal Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 A Geometric Approach to Maximizing the Circuit Depth that Can Be Evaluated 70
7.4 Instantiating the Ring: The Geometry of Polynomial Rings . . . . . . . . . 72
7.5 Instantiating Encrypt and Minimizing rEnc . . . . . . . . . . . . . . . . . . . 75
7.6 Instantiating Decrypt and Maximizing rDec . . . . . . . . . . . . . . . . . . . 75
7.7 Security of the Concrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . 77
7.8 How Useful is the Somewhat Homomorphic Scheme By Itself? . . . . . . . . 79
8 Tweaks to the Somewhat Homomorphic Scheme 81
8.1 On the Relationship between the Dual and the Inverse of an Ideal Lattice . 82
8.2 Transference Lemmas for Ideal Lattices . . . . . . . . . . . . . . . . . . . . 85
8.3 Tweaking the Decryption Equation . . . . . . . . . . . . . . . . . . . . . . . 86
8.4 A Tweak to Reduce the Circuit Complexity of the Rounding Step in Decryption 88
9 Decryption Complexity of the Tweaked Scheme 90
10 Squashing the Decryption Circuit 98
10.1 A Generic Description of the Transformation . . . . . . . . . . . . . . . . . 98
10.2 How to Squash, Concretely . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.3 Bootstrapping Achieved: The Decryption Circuit for the Transformed System 102
11 Security 104
11.1 Regarding the Hint Given in Our “Squashing” Transformation . . . . . . . 104
vii
11.2 Counterbalancing Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 113
12 Performance and Optimizations 115
12.1 Simple Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.2 Basic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.3 More Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13 Background on Ideal Lattices II 125
13.1 Overview of Gaussian Distributions over Lattices . . . . . . . . . . . . . . . 125
13.2 The Smoothing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
13.3 Sampling a Lattice According to a Gaussian Distribution . . . . . . . . . . 128
13.4 Ideal Factorization in Polynomial Rings . . . . . . . . . . . . . . . . . . . . 129
0/5000
Giới thiệu 11.1 một tổng quan rất ngắn gọn và không chính thức của chúng tôi xây dựng.......... 21.2 những gì là hoàn toàn mã hoá Homomorphic? . . . . . . . . . . . . . . . . . . . 51.3 bootstrapping một chương trình mà có thể đánh giá mạch giải mã riêng của nó. . 71.4 lưới lý tưởng: Lý tưởng để xây dựng Bootstrappable mã hóa... 101.5 squashing mạch giải mã: Encrypter bắt đầu giải mã! . . . . 151.6 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 các định nghĩa liên quan đến Homomorphic mã hóa 272.1 cơ bản định nghĩa................................. 272.2 tính toán an ninh định nghĩa...................... . 313 trước Homomorphic mã hóa các đề án 34Mã hóa bootstrappable 4 434.1 San lấp đầy đủ Homomorphic mã hóa từ mã hóa Bootstrappable, quát 434.2 tính đúng đắn, tính toán phức tạp và an ninh của bộ xây dựng chung 484.3 đầy đủ Homomorphic mã hóa từ KDM bảo mật mã hóa Bootstrappable 514.4 các mã hóa đầy đủ Homomorphic từ Bootstrappable mã hóa trong mô hình ngẫu nhiên Oracle 53vi5 một chương trình tóm tắt dựa trên vấn đề lý tưởng Coset 575.1 vấn đề lý tưởng Coset............................ 585.2 An Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Security of the Abstract Scheme . . . . . . . . . . . . . . . . . . . . . . . . 626 Background on Ideal Lattices I: The Basics 636.1 Basic Background on Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 636.2 Basic Background on Ideal Lattices . . . . . . . . . . . . . . . . . . . . . . . 656.3 Probability Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 A Somewhat Homomorphic Encryption Scheme 697.1 Why Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Why Ideal Lattices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.3 A Geometric Approach to Maximizing the Circuit Depth that Can Be Evaluated 707.4 Instantiating the Ring: The Geometry of Polynomial Rings . . . . . . . . . 727.5 Instantiating Encrypt and Minimizing rEnc . . . . . . . . . . . . . . . . . . . 757.6 Instantiating Decrypt and Maximizing rDec . . . . . . . . . . . . . . . . . . . 757.7 Security of the Concrete Scheme . . . . . . . . . . . . . . . . . . . . . . . . 777.8 How Useful is the Somewhat Homomorphic Scheme By Itself? . . . . . . . . 798 Tweaks to the Somewhat Homomorphic Scheme 818.1 On the Relationship between the Dual and the Inverse of an Ideal Lattice . 828.2 Transference Lemmas for Ideal Lattices . . . . . . . . . . . . . . . . . . . . 858.3 Tweaking the Decryption Equation . . . . . . . . . . . . . . . . . . . . . . . 86
8.4 A Tweak to Reduce the Circuit Complexity of the Rounding Step in Decryption 88
9 Decryption Complexity of the Tweaked Scheme 90
10 Squashing the Decryption Circuit 98
10.1 A Generic Description of the Transformation . . . . . . . . . . . . . . . . . 98
10.2 How to Squash, Concretely . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.3 Bootstrapping Achieved: The Decryption Circuit for the Transformed System 102
11 Security 104
11.1 Regarding the Hint Given in Our “Squashing” Transformation . . . . . . . 104
vii
11.2 Counterbalancing Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 113
12 Performance and Optimizations 115
12.1 Simple Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.2 Basic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.3 More Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13 Background on Ideal Lattices II 125
13.1 Overview of Gaussian Distributions over Lattices . . . . . . . . . . . . . . . 125
13.2 The Smoothing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
13.3 Sampling a Lattice According to a Gaussian Distribution . . . . . . . . . . 128
13.4 Ideal Factorization in Polynomial Rings . . . . . . . . . . . . . . . . . . . . 129
8.4 A Tweak to Reduce the Circuit Complexity of the Rounding Step in Decryption 88
9 Decryption Complexity of the Tweaked Scheme 90
10 Squashing the Decryption Circuit 98
10.1 A Generic Description of the Transformation . . . . . . . . . . . . . . . . . 98
10.2 How to Squash, Concretely . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10.3 Bootstrapping Achieved: The Decryption Circuit for the Transformed System 102
11 Security 104
11.1 Regarding the Hint Given in Our “Squashing” Transformation . . . . . . . 104
vii
11.2 Counterbalancing Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 113
12 Performance and Optimizations 115
12.1 Simple Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
12.2 Basic Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.3 More Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13 Background on Ideal Lattices II 125
13.1 Overview of Gaussian Distributions over Lattices . . . . . . . . . . . . . . . 125
13.2 The Smoothing Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
13.3 Sampling a Lattice According to a Gaussian Distribution . . . . . . . . . . 128
13.4 Ideal Factorization in Polynomial Rings . . . . . . . . . . . . . . . . . . . . 129
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