학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2022. 8. 이훈희.The advent of a quantum mechanical computer presents a clear threat to existing cryptography. On the other hand, the quantum computer also suggests the possibility of a new cryptographic protocol through the properties of quantum mechanics. These two perspectives, respectively, gave rise to a new field called post-quantum cryptography as a countermeasure against quantum attacks and quantum cryptography as a new cryptographic technology using quantum mechanics, which are the subject of this thesis.
In this thesis, we reconsider the security of the current post-quantum cryptography through a new quantum attack, model, and security proof. We present the fine-grained quantum security of hash functions as cryptographic primitives against preprocessing adversaries. We also bring recent quantum information theoretic research into cryptography, creating new quantum public key encryption and quantum commitment. Along the way, we resolve various open problems such as limitations of quantum algorithms with preprocessing computation, oracle separation problems in quantum complexity theory, and public key encryption using group action.양자역학을 이용한 컴퓨터의 등장은 쇼어의 알고리즘 등을 통해 기존 암호학에 명백한 위협을 제시하며, 양자역학의 성질을 통한 새로운 암호프로토콜의 가능성 또한 제시한다. 이러한 두 가지 관점은 각각 이 학위 논문의 주제가 되는 양자공격에 대한 대응책으로써의 대양자암호와 양자역학을 이용한 암호기술인 양자암호라고 불리는 새로운 분야를 발생시켰다.
이 학위 논문에서는 현재 대양자암호의 안전성을 새로운 양자암호 공격 알고리즘과 모델, 안전성 증명을 통해 재고한다. 특히 암호학적 해쉬함수의 일방향함수, 암호학적 의사난수생성기로서의 대양자 암호 안전성의 구체적인 평가를 제시한다. 또한 최근 양자역학의 연구를 양자암호에 도입함으로써 새로운 양자 공개키암호와 양자 커밋먼트 등의 새로운 발견을 제시한다. 이 과정에서 전처리 계산을 포함한 양자알고리즘의 한계, 양자 복잡계들의 오라클분리 문제, 군의 작용을 이용한 공개키 암호 등의 여러 열린문제들의 해결을 제시한다.1 Introduction 1
1.1 Contributions 3
1.2 Related Works 11
1.3 Research Papers 13
2 Preliminaries 14
2.1 Quantum Computations 15
2.2 Quantum Algorithms 20
2.3 Cryptographic Primitives 21
I Post-Quantum Cryptography: Attacks, New Models, and Proofs 24
3 Quantum Cryptanalysis 25
3.1 Introduction 25
3.2 QROM-AI Algorithm for Function Inversion 26
3.3 Quantum Multiple Discrete Logarithm Problem 34
3.4 Discussion and Open problems 39
4 Quantum Random Oracle Model with Classical Advice 42
4.1 Quantum ROM with Auxiliary Input 44
4.2 Function Inversion 46
4.3 Pseudorandom Generators 56
4.4 Post-quantum Primitives 58
4.5 Discussion and Open Problems 59
5 Quantum Random Permutations with Quantum Advice 62
5.1 Bound for Inverting Random Permutations 64
5.2 Preparation 64
5.3 Proof of Theorem 68
5.4 Implication in Complexity Theory 74
5.5 Discussion and Open Problems 77
II Quantum Cryptography: Public-key Encryptions and Bit Commitments 79
6 Equivalence Theorem 80
6.1 Equivalence Theorem 81
6.2 Non-uniform Equivalence Theorem 83
6.3 Proof of Equivalence Theorem 86
7 Quantum Public Key Encryption 89
7.1 Swap-trapdoor Function Pairs 90
7.2 Quantum-Ciphertext Public Key Encryption 94
7.3 Group Action based Construction 99
7.4 Lattice based Construction 107
7.5 Discussion and Open Problems 113
7.6 Deferred Proof 114
8 Quantum Bit Commitment 119
8.1 Quantum Commitments 120
8.2 Efficient Conversion 123
8.3 Applications of Conversion 126
8.4 Discussion and Open Problems 137박
