In response to the evolving landscape of quantum computing and the escalating vulnerabilities in classical cryptographic systems, our paper introduces a unified cryptographic framework. Rooted in the innovative work of Kuang et al., we leverage two novel primitives: the Quantum Permutation Pad (QPP) for symmetric key encryption and the Homomorphic Polynomial Public Key (HPPK) for Key Encapsulation Mechanism (KEM) and Digital Signatures (DS). Our approach adeptly confronts the challenges posed by quantum advancements. Utilizing the Galois Permutation Group's matrix representations and inheriting its bijective and non-commutative properties, QPP achieves quantum-secure symmetric key encryption, seamlessly extending Shannon's perfect secrecy to both classical and quantum-native systems. Meanwhile, HPPK, free from NP-hard problems, fortifies symmetric encryption for the plain public key. It accomplishes this by concealing the mathematical structure through modular multiplications or arithmetic representations of Galois Permutation Group over hidden rings, harnessing their partial homomorphic properties. This allows for secure computation on encrypted data during secret encapsulations, bolstering the security of the plain public key. The seamless integration of KEM and DS within HPPK cryptography yields compact key, cipher, and signature sizes, demonstrating exceptional performance. This paper organically unifies QPP and HPPK under the Galois Permutation Group, marking a significant advancement in laying the groundwork for quantum-resistant cryptographic protocols. Our contribution propels the development of secure communication systems amid the era of quantum computing.

翻译：针对量子计算持续演进及经典密码系统日益加剧的脆弱性，本文提出一种统一的密码学框架。该框架植根于Kuang等人的创新性工作，利用两种新型原语：用于对称密钥加密的量子置换垫(QPP)和用于密钥封装机制(KEM)与数字签名(DS)的同态多项式公钥(HPPK)。我们的方法巧妙应对量子技术进展带来的挑战。通过利用伽罗瓦置换群的矩阵表示并继承其双射性与非交换性，QPP实现了量子安全的对称密钥加密，将香农完美保密性无缝扩展至经典系统与量子原生系统。同时，HPPK无需依赖NP困难问题，即可为明文公钥强化对称加密。它通过模乘运算或隐藏环上伽罗瓦置换群的算术表示来隐藏数学结构，并利用其部分同态属性，从而在密钥封装过程中实现加密数据的安全计算，增强了明文公钥的安全性。HPPK密码体系内KEM与DS的无缝集成产生了紧凑的密钥、密文和签名尺寸，展现出卓越性能。本文在伽罗瓦置换群框架下有机统一了QPP与HPPK，为抗量子密码协议奠定基础，标志着重要进展。我们的贡献推动了量子计算时代安全通信系统的发展。