Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if FS does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of FS and the condition number of FS. As a consequence, we give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are small, say polynomial in the size of FS. We apply our quantum algorithm for solving Boolean equations to the cryptanalysis of several important cryptosystems: the stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, and the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the condition numbers of the corresponding equation systems are large. This leads to a new criterion for designing cryptosystems that can against the attack of quantum computers: their corresponding equation systems must have large condition numbers

Algorithms on Ideal over Complex Multiplication order

We show in this paper that the Gentry-Szydlo algorithm for cyclotomic orders, previously revisited by Lenstra-Silverberg, can be extended to complex-multiplication (CM) orders, and even to a more general structure. This algorithm allows to test equality over the polarized ideal class group, and finds a generator of the polarized ideal in polynomial time. Also, the algorithm allows to solve the norm equation over CM orders and the recent reduction of principal ideals to the real suborder can also be performed in polynomial time. Furthermore, we can also compute in polynomial time a unit of an order of any number field given a (not very precise) approximation of it. Our description of the Gentry-Szydlo algorithm is different from the original and Lenstra- Silverberg's variant and we hope the simplifications made will allow a deeper understanding. Finally, we show that the well-known speed-up for enumeration and sieve algorithms for ideal lattices over power of two cyclotomics can be generalized to any number field with many roots of unity.Comment: Full version of a paper submitted to ANT

정보 보호 기계 학습의 암호학 기반 기술: 근사 동형 암호와 부호 기반 암호

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 전기·정보공학부, 2021. 2. 노종선.In this dissertation, three main contributions are given as; i) a protocol of privacy-preserving machine learning using network resources, ii) the development of approximate homomorphic encryption that achieves less error and high-precision bootstrapping algorithm without compromising performance and security, iii) the cryptanalysis and the modification of code-based cryptosystems: cryptanalysis on IKKR cryptosystem and modification of the pqsigRM, a digital signature scheme proposed to the post-quantum cryptography (PQC) standardization of National Institute of Standards and Technology (NIST). The recent development of machine learning, cloud computing, and blockchain raises a new privacy problem; how can one outsource computation on confidential data? Moreover, as research on quantum computers shows success, the need for PQC is also emerging. Multi-party computation (MPC) is the cryptographic protocol that makes computation on data without revealing it. Since MPC is designed based on homomorphic encryption (HE) and PQC, research on designing efficient and safe HE and PQC is actively being conducted. First, I propose a protocol for privacy-preserving machine learning (PPML) that replaces bootstrapping of homomorphic encryption with network resources. In general, the HE ciphertext has a limited depth of circuit that can be calculated, called the level of a ciphertext. We call bootstrapping restoring the level of ciphertext that has exhausted its level through a method such as homomorphic decryption. Bootstrapping of homomorphic encryption is, in general, very expensive in time and space. However, when deep computations like deep learning are performed, it is required to do bootstrapping. In this protocol, both the client's message and servers' intermediate values are kept secure, while the client's computation and communication complexity are light. Second, I propose an improved bootstrapping algorithm for the CKKS scheme and a method to reduce the error by homomorphic operations in the CKKS scheme. The Cheon-Kim-Kim-Song (CKKS) scheme (Asiacrypt '17) is one of the highlighted fully homomorphic encryption (FHE) schemes as it is efficient to deal with encrypted real numbers, which are the usual data type for many applications such as machine learning. However, the precision drop due to the error growth is a drawback of the CKKS scheme for data processing. I propose a method to achieve high-precision approximate FHE using the following two methods .First, I apply the signal-to-noise ratio (SNR) concept and propose methods to maximize SNR by reordering homomorphic operations in the CKKS scheme. For that, the error variance is minimized instead of the upper bound of error when we deal with the encrypted data. Second, from the same perspective of minimizing error variance, I propose a new method to find the approximate polynomials for the CKKS scheme. The approximation method is especially applied to the CKKS scheme's bootstrapping, where we achieve bootstrapping with smaller error variance compared to the prior arts. In addition to the above variance-minimizing method, I cast the problem of finding an approximate polynomial for a modulus reduction into an L2-norm minimization problem. As a result, I find an approximate polynomial for the modulus reduction without using the sine function, which is the upper bound for the polynomial approximation of the modulus reduction. By using the proposed method, the constraint of q = O(m^{3/2}) is relaxed as O(m), and thus the level loss in bootstrapping can be reduced. The performance improvement by the proposed methods is verified by implementation over HE libraries, that is, HEAAN and SEAL. The implementation shows that by reordering homomorphic operations and using the proposed polynomial approximation, the reliability of the CKKS scheme is improved. Therefore, the quality of services of various applications using the proposed CKKS scheme, such as PPML, can be improved without compromising performance and security. Finally, I propose an improved code-based signature scheme and cryptanalysis of code-based cryptosystems. A novel code-based signature scheme with small parameters and an attack algorithm on recent code-based cryptosystems are presented in this dissertation. This scheme is based on a modified Reed-Muller (RM) code, which reduces the signing complexity and key size compared with existing code-based signature schemes. The proposed scheme has the advantage of the pqsigRM decoder and uses public codes that are more difficult to distinguish from random codes. I use (U, U+V) -codes with the high-dimensional hull to overcome the disadvantages of code-based schemes. The proposed a decoder which efficiently samples from coset elements with small Hamming weight for any given syndrome. The proposed signature scheme resists various known attacks on RM code-based cryptography. For 128 bits of classical security, the signature size is 4096 bits, and the public key size is less than 1 MB. Recently, Ivanov, Kabatiansky, Krouk, and Rumenko (IKKR) proposed three new variants of the McEliece cryptosystem (CBCrypto 2020, affiliated with Eurocrypt 2020). This dissertation shows that one of the IKKR cryptosystems is equal to the McEliece cryptosystem. Furthermore, a polynomial-time attack algorithm for the other two IKKR cryptosystems is proposed. The proposed attack algorithm utilizes the linearity of IKKR cryptosystems. Also, an implementation of the IKKR cryptosystems and the proposed attack is given. The proposed attack algorithm finds the plaintext within 0.2 sec, which is faster than the elapsed time for legitimate decryption.본 논문은 크게 다음의 세 가지의 기여를 포함한다. i) 네트워크를 활용해서 정보 보호 딥러닝을 개선하는 프로토콜 ii) 근사 동형 암호에서 보안성과 성능의 손해 없이 에러를 낮추고 높은 정확도로 부트스트래핑 하는 방법 iii) IKKR 암호 시스템과 pqsigRM 등 부호 기반 암호를 공격하는 방법과 효율적인 부호 기반 전자 서명 시스템. 근래의 기계학습과 블록체인 기술의 발전으로 인해서 기밀 데이터에 대한 연산을 어떻게 외주할 수 있느냐에 대한 새로운 보안 문제가 대두되고 있다. 또한, 양자 컴퓨터에 관한 연구가 성공을 거듭하면서, 이를 이용한 공격에 저항하는 포스트 양자 암호의 필요성 또한 커지고 있다. 다자간 컴퓨팅은 데이터를 공개하지 않고 데이터에 대한 연산을 수행할 수 있도록 하는 암호학적 프로토콜의 총칭이다. 다자간 컴퓨팅은 동형 암호와 포스트 양자 암호에 기반하고 있으므로, 효율적인 동형 암호와 포스트 양자 암호에 관한 연구가 활발하게 수행되고 있다. 동형 암호는 암호화된 데이터에 대한 연산이 가능한 특수한 암호화 알고리즘이다. 일반적으로 동형 암호의 암호문에 대해서 수행 가능한 연산의 깊이가 정해져 있으며, 이를 암호문의 레벨이라고 칭한다. 레벨을 모두 소비한 암호문의 레벨을 다시 복원하는 과정을 부트스트래핑 (bootstrapping)이라고 칭한다. 일반적으로 부트스트래핑은 매우 오래 걸리는 연산이며 시간 및 공간 복잡도가 크다. 그러나, 딥러닝과 같이 깊이가 큰 연산을 수행하는 경우 부트스트래핑이 필수적이다. 본 논문에서는 정보 보호 기계학습을 위한 새로운 프로토콜을 제안한다. 이 프로토콜에서는 입력 메시지와 더불어 신경망의 중간값들 또한 안전하게 보호된다. 그러나 여전히 사용자의 통신 및 연산 복잡도는 낮게 유지된다. Cheon, Kim, Kim 그리고 Song (CKKS)가 제안한 암호 시스템 (Asiacrypt 17)은 기계학습 등에서 가장 널리 쓰이는 데이터인 실수를 효율적으로 다룰 수 있으므로 가장 촉망받는 완전 동형 암호 시스템이다. 그러나, 오류의 증폭과 전파가 CKKS 암호 시스템의 가장 큰 단점이다. 이 논문에서는 아래의 기술을 활용하여 CKKS 암호 시스템의 오류를 줄이는 방법을 제안하며, 이는 근사 동형 암호에 일반화하여 적용할 수 있다. 첫째, 신호 대비 잡음 비 (signal-to-noise ratio, SNR)의 개념을 도입하여, SNR를 최대화하도록 연산의 순서를 재조정한다. 그러기 위해서는, 오류의 최대치 대신 분산이 최소화되어야 하며, 이를 관리해야 한다. 둘째, 오류의 분산을 최소화한다는 같은 관점에서 새로운 다항식 근사 방법을 제안한다. 이 근사 방법은 특히, CKKS 암호 시스템의 부트스트래핑에 적용되었으며, 종래 기술보다 더 낮은 오류를 달성한다. 위의 방법에 더하여, 근사 다항식을 구하는 문제를 L2-norm 최소화 문제로 치환하는 방법을 제안한다. 이를 통해서 사인 함수의 도입 없이 근사 다항식을 구하는 방법을 제안한다. 제안된 방법을 사용하면, q=O(m^{3/2})라는 제약을 q=O(m)으로 줄일 수 있으며, 부트스트래핑에 필요한 레벨 소모를 줄일 수 있다. 성능 향상은 HEAAN과 SEAL 등의 동형 암호 라이브러리를 활용한 구현을 통해 증명했으며, 구현을 통해서 연산 재정렬과 새로운 부트스트래핑이 CKKS 암호 시스템의 성능을 향상함을 확인했다. 따라서, 보안성과 성능의 타협 없이 근사 동형 암호를 사용하는 서비스의 질을 향상할 수 있다. 양자 컴퓨터를 활용하여 전통적인 공개키 암호를 공격하는 효율적인 알고리즘이 공개되면서, 포스트 양자 암호에 대한 필요성이 증대했다. 부호 기반 암호는 포스트 양자 암호로써 널리 연구되었다. 작은 키 크기를 갖는 새로운 부호 기반 전자 서명 시스템과 부호 기반 암호를 공격하는 방법이 논문에 제안되어 있다. pqsigRM이라 명명한 전자 서명 시스템이 그것이다. 이 전자 서명 시스템은 수정된 Reed-Muller (RM) 부호를 활용하며, 서명의 복잡도와 키 크기를 종래 기술보다 많이 줄인다. pqsigRM은 hull의 차원이 큰 (U, U+V) 부호와 이의 복호화를 이용하여, 서명에서 큰 이득이 있다. 이 복호화 알고리즘은 주어진 모든 코셋 (coset)의 원소에 대하여 작은 헤밍 무게를 갖는 원소를 반환한다. 또한, 수정된 RM 부호를 이용하여, 알려진 모든 공격에 저항한다. 128비트 안정성에 대해서 서명의 크기는 4096 비트이고, 공개 키의 크기는 1MB보다 작다. 최근, Ivanov, Kabatiansky, Krouk, 그리고 Rumenko (IKKR)가 McEliece 암호 시스템의 세 가지 변형을 발표했다 (CBCrypto 2020, Eurocrypt 2020와 함께 진행). 본 논문에서는 IKKR 암호 시스템중 하나가 McEliece 암호 시스템과 동치임을 증명한다. 또한 나머지 IKKR 암호 시스템에 대한 다항 시간 공격을 제안한다. 제안하는 공격은 IKKR 암호 시스템의 선형성을 활용한다. 또한, 이 논문은 제안한 공격의 구현을 포함하며, 제안된 공격은 0.2초 이내에 메시지를 복원하고, 이는 정상적인 복호화보다 빠른 속도이다.Contents Abstract i Contents iv List of Tables ix List of Figures xi 1 Introduction 1 1.1 Homomorphic Encryption and Privacy-Preserving Machine Learning 4 1.2 High-Precision CKKS Scheme and Its Bootstrapping 5 1.2.1 Near-Optimal Bootstrapping of the CKKS Scheme Using Least Squares Method 6 1.2.2 Variance-Minimizing and Optimal Bootstrapping of the CKKS Scheme 8 1.3 Efficient Code-Based Signature Scheme and Cryptanalysis of the Ivanov-Kabatiansky-Krouk-Rumenko Cryptosystems 10 1.3.1 Modified pqsigRM: An Efficient Code-Based Signature Scheme 11 1.3.2 Ivanov-Kabatiansky-Krouk-Rumenko Cryptosystems and Its Equality 13 1.4 Organization of the Dissertation 14 2 Preliminaries 15 2.1 Basic Notation 15 2.2 Privacy-Preserving Machine Learning and Security Terms 16 2.2.1 Privacy-Preserving Machine Learning and Security Terms 16 2.2.2 Privacy-Preserving Machine Learning 17 2.3 The CKKS Scheme and Its Bootstrapping 18 2.3.1 The CKKS Scheme 18 2.3.2 CKKS Scheme in RNS 22 2.3.3 Bootstrapping of the CKKS Scheme 24 2.3.4 Statistical Characteristics of Modulus Reduction and Failure Probability of Bootstrapping of the CKKS Scheme 26 2.4 Approximate Polynomial and Signal-to-Noise Perspective for Approximate Homomorphic Encryption 27 2.4.1 Chebyshev Polynomials 27 2.4.2 Signal-to-Noise Perspective of the CKKS Scheme 28 2.5 Preliminary for Code-Based Cryptography 29 2.5.1 The McEliece Cryptosystem 29 2.5.2 CFS Signature Scheme 30 2.5.3 ReedMuller Codes and Recursive Decoding 31 2.5.4 IKKR Cryptosystems 33 3 Privacy-Preserving Machine Learning via FHEWithout Bootstrapping 37 3.1 Introduction 37 3.2 Information Theoretic Secrecy and HE for Privacy-Preserving Machine Learning 38 3.2.1 The Failure Probability of Ordinary CKKS Bootstrapping 39 3.3 Comparison With Existing Methods 43 3.3.1 Comparison With the Hybrid Method 43 3.3.2 Comparison With FHE Method 44 3.4 Comparison for Evaluating Neural Network 45 4 High-Precision Approximate Homomorphic Encryption and Its Bootstrapping by Error Variance Minimization and Convex Optimization 50 4.1 Introduction 50 4.2 Optimization of Error Variance in the Encrypted Data 51 4.2.1 Tagged Information for Ciphertext 52 4.2.2 WorstCase Assumption 53 4.2.3 Error in Homomorphic Operations of the CKKS Scheme 54 4.2.4 Reordering Homomorphic Operations 59 4.3 Near-Optimal Polynomial for Modulus Reduction 66 4.3.1 Approximate Polynomial Using L2-Norm optimization 66 4.3.2 Efficient Homomorphic Evaluation of the Approximate Polynomial 70 4.4 Optimal Approximate Polynomial and Bootstrapping of the CKKS Scheme 73 4.4.1 Polynomial Basis Error and Polynomial Evaluation in the CKKS Scheme 73 4.4.2 Variance-Minimizing Polynomial Approximation 74 4.4.3 Optimal Approximate Polynomial for Bootstrapping and Magnitude of Its Coefficients 75 4.4.4 Reducing Complexity and Error Using Odd Function 79 4.4.5 Generalization of Weight Constants and Numerical Method 80 4.5 Comparison and Implementation 84 4.6 Reduction of Level Loss in Bootstrapping 89 4.7 Implementation of the Proposed Method and Performance Comparison 92 4.7.1 Error Variance Minimization 92 4.7.2 Weight Constant and Minimum Error Variance 93 4.7.3 Comparison of the Proposed MethodWith the Previous Methods 96 5 Efficient Code-Based Signature Scheme and Cryptanalysis of Code-Based Cryptosystems 104 5.1 Introduction 104 5.2 Modified ReedMuller Codes and Proposed Signature Scheme 105 5.2.1 Partial Permutation of Generator Matrix and Modified ReedMuller Codes 105 5.2.2 Decoding of Modified ReedMuller Codes 108 5.2.3 Proposed Signature Scheme 110 5.3 Security Analysis of Modified pqsigRM 111 5.3.1 Decoding One Out of Many 112 5.3.2 Security Against Key Substitution Attacks 114 5.3.3 EUFCMA Security 114 5.4 Indistinguishability of the Public Code and Signature 120 5.4.1 Modifications of Public Code 121 5.4.2 Public Code Indistinguishability 124 5.4.3 Signature Leaks 126 5.5 Parameter Selection 126 5.5.1 Parameter Sets 126 5.5.2 Statistical Analysis for Determining Number of Partial Permutations 128 5.6 Equivalence of the Prototype IKKR and the McEliece Cryptosystems 131 5.7 Cryptanalysis of the IKKR Cryptosystems 133 5.7.1 Linearity of Two Variants of IKKR Cryptosystems 133 5.7.2 The Attack Algorithm 134 5.7.3 Implementation 135 6 Conclusion 139 6.1 Privacy-Preserving Machine Learning Without Bootstrapping 139 6.2 Variance-Minimization in the CKKS Scheme 140 6.3 L2-Norm Minimization for the Bootstrapping of the CKKS Scheme 141 6.4 Modified pqsigRM: RM Code-Based Signature Scheme 142 6.5 Cryptanalysis of the IKKR Cryptosystem 143 Abstract (In Korean) 155 Acknowlegement 158Docto

Quantum Lightning Never Strikes the Same State Twice

Public key quantum money can be seen as a version of the quantum no-cloning theorem that holds even when the quantum states can be verified by the adversary. In this work, investigate quantum lightning, a formalization of "collision-free quantum money" defined by Lutomirski et al. [ICS'10], where no-cloning holds even when the adversary herself generates the quantum state to be cloned. We then study quantum money and quantum lightning, showing the following results: - We demonstrate the usefulness of quantum lightning by showing several potential applications, such as generating random strings with a proof of entropy, to completely decentralized cryptocurrency without a block-chain, where transactions is instant and local. - We give win-win results for quantum money/lightning, showing that either signatures/hash functions/commitment schemes meet very strong recently proposed notions of security, or they yield quantum money or lightning. - We construct quantum lightning under the assumed multi-collision resistance of random degree-2 systems of polynomials. - We show that instantiating the quantum money scheme of Aaronson and Christiano [STOC'12] with indistinguishability obfuscation that is secure against quantum computers yields a secure quantum money schem

Achievable secrecy enchancement through joint encryption and privacy amplification

In this dissertation we try to achieve secrecy enhancement in communications by resorting to both cryptographic and information theoretic secrecy tools and metrics. Our objective is to unify tools and measures from cryptography community with techniques and metrics from information theory community that are utilized to provide privacy and confidentiality in communication systems. For this purpose we adopt encryption techniques accompanied with privacy amplification tools in order to achieve secrecy goals that are determined based on information theoretic and cryptographic metrics. Every secrecy scheme relies on a certain advantage for legitimate users over adversaries viewed as an asymmetry in the system to deliver the required security for data transmission. In all of the proposed schemes in this dissertation, we resort to either inherently existing asymmetry in the system or proactively created advantage for legitimate users over a passive eavesdropper to further enhance secrecy of the communications. This advantage is manipulated by means of privacy amplification and encryption tools to achieve secrecy goals for the system evaluated based on information theoretic and cryptographic metrics. In our first work discussed in Chapter 2 and the third work explained in Chapter 4, we rely on a proactively established advantage for legitimate users based on eavesdropper’s lack of knowledge about a shared source of data. Unlike these works that assume an errorfree physical channel, in the second work discussed in Chapter 3 correlated erasure wiretap channel model is considered. This work relies on a passive and internally existing advantage for legitimate users that is built upon statistical and partial independence of eavesdropper’s channel errors from the errors in the main channel. We arrive at this secrecy advantage for legitimate users by exploitation of an authenticated but insecure feedback channel. From the perspective of the utilized tools, the first work discussed in Chapter 2 considers a specific scenario where secrecy enhancement of a particular block cipher called Data Encryption standard (DES) operating in cipher feedback mode (CFB) is studied. This secrecy enhancement is achieved by means of deliberate noise injection and wiretap channel encoding as a technique for privacy amplification against a resource constrained eavesdropper. Compared to the first work, the third work considers a more general framework in terms of both metrics and secrecy tools. This work studies secrecy enhancement of a general cipher based on universal hashing as a privacy amplification technique against an unbounded adversary. In this work, we have achieved the goal of exponential secrecy where information leakage to adversary, that is assessed in terms of mutual information as an information theoretic measure and Eve’s distinguishability as a cryptographic metric, decays at an exponential rate. In the second work generally encrypted data frames are transmitted through Automatic Repeat reQuest (ARQ) protocol to generate a common random source between legitimate users that later on is transformed into information theoretically secure keys for encryption by means of privacy amplification based on universal hashing. Towards the end, future works as an extension of the accomplished research in this dissertation are outlined. Proofs of major theorems and lemmas are presented in the Appendix