  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.
