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：针对量子计算技术的持续演进以及经典密码系统日益凸显的脆弱性，本文提出了一种统一的密码框架。该框架根植于Kuang等人的创新工作，利用两种新型原语：用于对称密钥加密的量子置换垫（QPP）以及用于密钥封装机制（KEM）和数字签名（DS）的同态多项式公钥（HPPK）。我们的方法有效应对了量子技术发展带来的挑战。通过利用伽罗瓦置换群的矩阵表示并继承其双射与非交换性质，QPP实现了量子安全的对称密钥加密，将香农的完美保密性无缝扩展至经典系统与量子原生系统。与此同时，HPPK无需依赖NP难问题，通过模乘运算或伽罗瓦置换群在隐藏环上的算术表示来隐藏数学结构，利用其部分同态性质，从而增强明文公钥的安全性。这使得在秘密封装过程中能够对加密数据执行安全计算，强化了明文公钥的安全保障。HPPK密码学中KEM与DS的有机整合，生成了紧凑的密钥、密文和签名尺寸，展现出卓越的性能。本文在伽罗瓦置换群框架下将QPP与HPPK统一整合，为构建抗量子密码协议奠定了关键基础。我们的贡献将推动量子计算时代安全通信系统的发展。