Mathematical analysis of the indistinguishability obfuscations

다국어 초록 (Multilingual Abstract)

Indistinguishability obfuscation (iO) is a weak notion of the program obfuscation which requires that if two functionally equivalent circuits are given, their obfuscated programs are indistinguishable. The existence of iO implies numerous cryptographic primitives such as multilinear map, functional encryption, non interactive multi-party key exchange. In gen- eral, many iO schemes are based on branching programs, and candidates of multilinear maps represented by GGH13, CLT13 and GGH15.

In this thesis, we present cryptanalyses of branching program based iO over multilinear maps GGH13 and GGH15. First, we propose cryptanaly- ses of all existing branching program based iO schemes over GGH13 for all recommended parameter settings. To achieve this, we introduce two novel techniques, ‘program converting’ using NTRU-solver and ‘matrix zeroiz- ing’, which can be applied to a wide range of obfuscation constructions. We then show that there exists polynomial time reduction from the NTRU problem to all known branching program based iO over GGH13.

Moreover, we propose a new attack on iO based on GGH15 which exploits statistical properties rather than algebraic approaches. We apply our attack to recent two obfuscations called CVW and BGMZ obfuscations. Thus, we break the CVW obfuscation under the current parameter setup, and show that algebraic security model of BGMZ obfuscation is not enough to achieve ideal security. We show that our attack is lying outside of the algebraic security model by presenting some parameters not captured by the proof of the model.

국문 초록 (Abstract)

기능성이 같은 두 프로그램과, 그 난독화된 프로그램들이 있을 때, 난독화된 프로그 램들을 구분할 수 없다면 구분불가능한 난독화라고 한다. 구분불가능한 난독화가 존재한다면, 다중선형함수, 함수암호, 다자간 키교환 등 많은 암호학적인 응용들이 존재하기 때문에, 구분불가능한 난독화를 설계하는 것은 매우 중요한 문제 중 하나 이다. 일반적으로, 많은 구분불가능한 난독화들은 다중선형함수 GGH13, CLT13, GGH15를 기반으로 하여 설계되었다.
본 학위 논문에서는, 다중선형함수를 기반으로 하는 난독화 기술들에 대한 안 전성 분석을 진행한다. 먼저, GGH13 다중선형함수를 기반으로 하는 모든 난독화 기술들은 현재 파라미터 하에 안전하지 않음을 보인다. 프로그램 변환(program converting), 행렬 제로화 공격(matrix zeroizing attack)이라는 두 가지 새로운 방 법을 제안하여 안전성을 분석하였고, 그 결과, 현존하는 모든 GGH13 다중선형함수 기반 난독화 기술이 다항식 시간 내에 NTRU 문제로 환원됨을 보인다.
또한, GGH15 다중선형함수를 기반으로 하는 난독화 기술에 대한 통계적인 공격방법을 제안한다. 통계적 공격방법을 최신 기술인 CVW 난독화, BGMZ 난독 화에 적용하여, CVW 난독화가 현재 파라미터에서 안전하지 않음을 보인다. 또한 BGMZ 난독화에서 제안한 대수적 안전성 모델이 이상적인 난독화 기술을 설계하 는데 충분하지 않다는 것을 보인다. 실제로, BGMZ 난독화가 안전하지 않은 특이한 파라미터를 제안하여, 우리 공격이 BGMZ에서 제안한 안전성 모델에 해당하지 않 음을 보인다.

목차 (Table of Contents)

• 1. Introduction 1
• 1.1 Indistinguishability Obfuscation 1
• 1.2 Contributions 4
• 1.2.1 Mathematical Analysis of iO based on GGH13 4
• 1.2.2 Mathematical Analysis of iO based on GGH15 5
• 1.3 List of Papers 6
• 2 Preliminaries 7
• 2.1 Basic Notations 7
• 2.2 Indistinguishability Obfuscation 8
• 2.3 Cryptographic Multilinear Map 9
• 2.4 Matrix Branching Program 10
• 2.5 Tensor product and vectorization . 11
• 2.6 Background Lattices . 12
• 3 Mathematical Analysis of Indistinguishability Obfuscation based on the GGH13 Multilinear Map 13
• 3.1 Preliminaries 14
• 3.1.1 Notations 14
• 3.1.2 GGH13 Multilinear Map 14
• 3.2 Main Theorem 17
• 3.3 Attackable BP Obfuscations 18
• 3.3.1 Randomization for Attackable Obfuscation Model 20
• 3.3.2 Encoding by Multilinear Map 21
• 3.3.3 Linear Relationally Inequivalent Branching Programs 22
• 3.4 Program Converting Technique 23
• 3.4.1 Converting to R Program 24
• 3.4.2 Recovering <g> and Converting to R/<g> Program 27
• 3.4.3 Analysis of the Converting Technique 28
• 3.5 Matrix Zeroizing Attack 29
• 3.5.1 Existing BP Obfuscations 31
• 3.5.2 Attackable BP Obfuscation, General Case 34
• 4 Mathematical Analysis of Indistinguishability Obfuscation based on the GGH15 Multilinear Map 37
• 4.1 Preliminaries 38
• 4.1.1 Notations 38
• 4.2 Statistical Zeroizing Attack . 39
• 4.2.1 Distinguishing Distributions using Sample Variance 42
• 4.3 Cryptanalysis of CVW Obfuscation 44
• 4.3.1 Construction of CVW Obfuscation 45
• 4.3.2 Cryptanalysis of CVW Obfuscation 48
• 4.4 Cryptanalysis of BGMZ Obfuscation 56
• 4.4.1 Construction of BGMZ Obfuscation 56
• 4.4.2 Cryptanalysis of BGMZ Obfuscation 59
• 5 Conclusions 65
• 6 Appendix 66
• 6.1 Appendix of Chapter 3 66
• 6.1.1 Extended Attackable Model 66
• 6.1.2 Examples of Matrix Zeroizing Attack 68
• 6.1.3 Examples of Linear Relationally Inequivalent BPs 70
• 6.1.4 Read-once BPs from NFA 70
• 6.1.5 Input-unpartitionable BPs from Barringtons Theorem 71
• 6.2 Appendix of Chapter 5 73
• 6.2.1 Simple GGH15 obfuscation 73
• 6.2.2 Modified CVW Obfuscation . 75
• 6.2.3 Transformation of Branching Programs 76
• 6.2.4 Modification of CVW Obfuscation 77
• 6.2.5 Assumptions of lattice preimage sampling 78
• 6.2.6 Useful Tools for Computing the Variances 79
• 6.2.7 Analysis of CVW Obfuscation 84
• 6.2.8 Analysis of BGMZ Obfuscation 97
• Abstract (in Korean) 117