摘 要：几何代数作为一种融合了几何与代数特性的数学工具，近年来在密码学领域展现出重要潜力，其多维向量空间的表达能力和运算特性为解决复杂密码问题提供了新思路本研究旨在探讨几何代数在密码学中的应用进展，通过分析其理论框架与实际算法设计，提出基于几何代数的新型密码构造方法研究发现，几何代数能够有效描述和优化非对称加密、数字签名及同态加密等核心密码技术，尤其在提升算法效率和抗量子攻击能力方面具有显著优势创新点在于首次将几何代数的旋量理论引入密钥生成机制，并通过实验验证了该方法在高维数据加密场景下的优越性能结果表明，基于几何代数的密码方案不仅具备更高的安全性，还能显著降低计算开销，为后量子密码学的发展提供了重要参考结论认为，几何代数有望成为未来密码学研究的核心工具之一，值得进一步深入探索其理论边界与工程实现
关键词：几何代数；密码学；旋量理论；后量子密码学；高维数据加密
Advances in the Application of Geometric Algebra in Cryptography
英文人名
Directive teacher:×××
Abstract：Geometric algebra, as a mathematical tool that integrates geometric and algebraic characteristics, has demonstrated significant potential in the field of cryptography in recent years. Its ability to express multidimensional vector spaces and its operational properties offer new approaches to solving complex cryptographic problems. This study investigates the advancements of geometric algebra applications in cryptography by analyzing its theoretical fr amework and practical algorithm design, proposing novel cryptographic construction methods based on geometric algebra. The findings reveal that geometric algebra can effectively describe and optimize key cryptographic technologies such as asymmetric encryption, digital signatures, and homomorphic encryption, particularly showing substantial advantages in enhancing algorithm efficiency and resistance to quantum attacks. An innovation lies in the first introduction of the spinor theory of geometric algebra into key generation mechanisms, with experimental validation confirming superior performance in high-dimensional data encryption scenarios. The results indicate that cryptographic schemes based on geometric algebra not only provide higher security but also significantly reduce computational overhead, offering crucial references for the development of post-quantum cryptography. It is concluded that geometric algebra holds promise as one of the core tools in future cryptographic research, warranting further exploration of its theoretical boundaries and engineering implementation.
Keywords: Geometric Algebra；Cryptography；Spinor Theory；Post-Quantum Cryptography；High-Dimensional Data Encryption
目 录
引言 1
一、几何代数基础与密码学关联 1
(一)几何代数的基本概念 1
(二)密码学中的数学需求 2
(三)几何代数在密码学中的初步应用 2
二、几何代数在对称加密中的应用进展 3
(一)对称加密算法的几何特性分析 3
(二)基于几何代数的密钥生成机制 3
(三)几何代数优化对称加密效率的研究 4
三、几何代数在公钥密码中的应用探索 4
(一)公钥密码体系的几何代数建模 4
(二)几何代数在椭圆曲线密码中的应用 5
(三)提高公钥密码安全性的几何方法 5
四、几何代数在后量子密码中的潜力分析 6
(一)后量子密码的几何代数框架构建 6
(二)几何代数对抗量子攻击的能力评估 6
(三)未来发展方向与技术挑战 7
结论 7
参考文献 9
致谢 9
