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Paper 2024/174

QPP and HPPK: Unifying Non-Commutativity for Quantum-Secure Cryptography with Galois Permutation Group

Abstract

In response to the evolving landscape of quantum computing and the heightened vulnerabilities in classical cryptographic systems, our paper introduces a comprehensive cryptographic framework. Building upon the pioneering work of Kuang et al., we present a unification of two innovative 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). By harnessing matrix representations of the Galois Permutation Group 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. Simultaneously, HPPK, free of NP-hard problems, relies on the security of symmetric encryption for the plain public key. This is accomplished by concealing the mathematical structure through arithmetic representations or modular multiplicative operators (arithmetic QPP) of the Galois Permutation Group over hidden rings, utilizing their partial homomorphic properties. This ensures secure computation on encrypted data during secret encapsulations, thereby enhancing the security of the plain public key. The integration of KEM and DS within HPPK cryptography results in compact key, cipher, and signature sizes, showcasing exceptional performance. This paper organically unifies QPP and HPPK under the Galois Permutation Group, marking a significant advance in laying the groundwork for quantum-resistant cryptographic protocols. Our contribution propels the development of secure communication systems in the era of quantum computing.