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

TL;DR

The paper addresses quantum-era cryptographic vulnerabilities by unifying symmetric and asymmetric approaches under the Galois Permutation Group. It introduces Quantum Permutation Pad (QPP) for quantum-secure symmetric encryption and Homomorphic Polynomial Public Key (HPPK) for KEM and Digital Signatures, leveraging matrix and arithmetic representations to achieve non-commutative, quantum-resistant operations. QPP extends Shannon's perfect secrecy to quantum contexts via random permutation spaces, while HPPK uses hidden-ring modular multiplicative permutations to realize compact, homomorphic KEM/DS without NP-hard assumptions. The integrated key triple enables combined encapsulation and signing workflows with compact key/cipher/signature sizes, offering a practical pathway toward quantum-resistant secure communications. The work emphasizes the operational potential across quantum-classical and quantum-native channels and lays groundwork for future protocol developments in quantum-secure infrastructures.

Abstract

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.