摘要
量子计算作为新兴的计算范式，其独特的并行处理能力为密码学带来了前所未有的挑战与机遇。本文旨在探讨量子计算的基本原理及其在密码学领域的潜在应用价值，通过分析量子比特、量子纠缠及量子门等核心概念，揭示量子算法对传统加密体系的影响。研究采用理论推导与案例分析相结合的方法，重点考察Shor算法和Grover算法对公钥密码体制和对称密钥密码体制的安全性冲击。研究表明，Shor算法能够在多项式时间内破解基于大整数分解和离散对数问题的传统公钥加密系统，而Grover算法则将穷举搜索的时间复杂度从O(2^n)降低至O(2^(n/2))，显著削弱了对称密钥系统的安全性。针对上述威胁，本文提出构建抗量子攻击的新型密码体制，包括基于格理论、多变量多项式和编码理论的后量子密码方案。研究结果表明，这些新体制具有良好的抗量子攻击特性，能够有效抵御量子计算机带来的安全风险。本研究不仅为理解量子计算与密码学的交互作用提供了理论依据，还为未来信息安全技术的发展指明了方向，特别是为设计适应量子时代的加密算法奠定了基础。
关键词：量子计算；密码学；Shor算法；Grover算法；后量子密码
Abstract
Quantum computing, as an emerging computational paradigm, presents unprecedented challenges and opportunities for cryptography due to its unique parallel processing capabilities. This paper aims to explore the fundamental principles of quantum computing and its potential applications in the field of cryptography by analyzing core concepts such as qubits, quantum entanglement, and quantum gates, thereby revealing the impact of quantum algorithms on traditional encryption systems. The study employs a combination of theoretical derivation and case analysis, focusing on the security implications of Shor's algorithm and Grover's algorithm on public-key and symmetric-key cryptographic systems. Research findings indicate that Shor's algorithm can factor large integers and solve discrete logarithm problems in polynomial time, thereby compromising the security of traditional public-key cryptosystems. Meanwhile, Grover's algorithm reduces the time complexity of exhaustive search from O(2^n) to O(2^(n/2)), significantly weakening the security of symmetric-key systems. In response to these threats, this paper proposes constructing quantum-resistant cryptographic systems, including post-quantum cryptography schemes based on lattice theory, multivariate polynomials, and coding theory. The results show that these new systems possess robust resistance against quantum attacks and can effectively mitigate the security risks posed by quantum computers. This research not only provides a theoretical basis for understanding the interaction between quantum computing and cryptography but also points out the direction for the development of future information security technologies, particularly in designing encryption algorithms adapted to the quantum era.
Keywords：Quantum Computing; Cryptography; Shor Algorithm; Grover Algorithm; Post-Quantum Cryptography
目 录
摘要 I
Abstract II
一、绪论 1
(一) 研究背景与意义 1
(二) 国内外研究现状 1
二、量子计算基础原理 2
(一) 量子比特与量子态 2
(二) 量子门与量子电路 2
(三) 量子算法及其特性 3
(四) 量子计算的优越性分析 4
三、密码学中的量子威胁 4
(一) 传统密码体制的脆弱性 4
(二) Shor算法对公钥密码的影响 5
(三) Grover算法对对称密码的影响 5
(四) 量子攻击的具体案例分析 6
四、量子密码学的发展方向 6
(一) 量子密钥分发协议 6
(二) 量子安全直接通信 7
(三) 后量子密码体制构建 8
(四) 量子密码的实际应用前景 8
结 论 10
参考文献 11
