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.
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.
This review delves into the latest advancements in quantum-secure cryptography, focusing on the quantum permutation pad (QPP), a pivotal innovation proposed by Kuang et al. QPP harnesses the non-commutativity and generalized uncertainty derived from the Galois permutation group, making it highly suitable for cryptographic applications. The review underscores QPP’s versatility across both symmetric and asymmetric cryptography through three core representations: matrix-based for classical encryption, quantum gates for quantum-native encryption, and arithmetic-based for multivariate public key systems such as Merkle–Hellman cryptosystems, multivariate public key cryptography (MPKC), and the most recent homomorphic polynomial public key (HPPK). In particular, QPP strengthens the security of HPPK’s key encapsulation mechanism (KEM) and digital signature (DS) schemes, thus offering robust quantum resistance. This work further examines QPP’s integration with various encryption techniques for enhancing resilience against quantum attacks. By addressing challenges in cryptographic complexity, key size optimization, and security enhancement, the review presents a thorough evaluation of QPP’s role in fortifying cryptographic protocols for ensuring strong security foundations in the quantum computing era.
Abstract Quantum permutation pad or QPP is a quantum-safe symmetric cryptographic algorithm proposed by Kuang and Bettenburg in 2020. The theoretical foundation of QPP leverages the linear algebraic representations of quantum gates which makes QPP realizable in both, quantum and classical systems. By applying the QPP with 64 of 8-bit permutation gates, holding respective entropy of over 100,000 bits, we accomplished quantum random number distributions digitally over today’s classical internet. The QPP has also been used to create pseudo quantum random numbers and served as a foundation for quantum-safe lightweight block and streaming ciphers. This paper continues to explore numerous applications of QPP, namely, we present an implementation of QPP as a quantum encryption circuit on today’s still noisy quantum computers. With the publicly available 5-qubit IBMQ devices, we demonstrate quantum secure encryption (256 bits of entropy) using 2-qubit QPP with 56 permutation gates, and 3-qubit QPP with 17 permutation gates respectively. Initial qubits of the encryption circuit correspond to the plaintext and after applying quantum encryption operations, cipher qubits are measured with probabilistic distributions, and the results with the highest probability are recorded as cipher bits. The cipher bits are then decrypted with an inverse QPP circuit. The output state plaintext qubits are measured and the most frequent count measurement results are recorded as plaintext bits. This quantum encryption and decryption process clearly demonstrates that QPP quantum implementations works exactly as symmetric encryption and decryption schemes should. The plaintext and ciphertext bits can also be encrypted and decrypted respectively by any classical computing device with the corresponding QPP algorithm as in quantum computers. This work reveals that it is possible to build quantum-secure communications between quantum-to-quantum and quantum-to-classical computers over today’s internet and the future quantum internet.
This paper introduces Quantum Cryptographic Dynamics (QCD), a novel theoretical framework that models cryptographic processes through the lens of entropy injection and ejection. Drawing inspiration from classical mechanics, QCD establishes three fundamental laws: the Entropy Inertial Law (conservation of entropy in isolated systems), the Entropy Evolution Law (transformation of entropy via injection operators), and the Entropy Redistribution Law (reversibility through ejection operators). Applying these principles, we provide a unified, entropy-centric interpretation of classical and quantum cryptographic schemes, including symmetric-key systems, public-key protocols, and post-quantum cryptography (PQC) algorithms such as Learning With Errors (LWE), Kyber, and Homomorphic Polynomial Public Key (HPPK). By shifting the focus from computational hardness assumptions to the fundamental dynamics of entropy manipulation, QCD offers new insights into the security foundations of these cryptographic primitives. Furthermore, we reinterpret the Quantum Permutation Pad Random Number Generator (QPP-RNG) within the QCD framework. QPP-RNG is modeled as an entropy-driven process that harnesses system jitter to generate unpredictable random numbers, which in turn fuel PQC schemes like Kyber and HPPK for quantum-secure key establishment, forming a self-sustained quantum-secure eco-cryptosystem. This framework provides a rigorous approach to security analysis, unifying cryptographic security models across classical, quantum, and post-quantum domains. QCD establishes a robust foundation for designing quantum-resistant cryptographic primitives and assessing the fundamental security properties of cryptographic systems.
Quantum permutation pad or QPP is a set of quantum permutation gates. QPP has been demonstrated for quantum secure encryption in both classical and quantum computing systems recently, even at a noisy 5-qubit IBMQ systems. In a classical computing system, QPP encryption is implemented as a permutation gate matrix multiplication with information state vectors. In a quantum computing system, QPP is compiled into a quantum encryption circuit in a native quantum computer and encryption is performed through QPP circuit. Leveraging its quantum mechanical characteristics, we report a digital QKD or D-QKD platform using QPP as a quantum mechanical algorithm implemented in classical systems to distribute quantum entropy, generated from physical quantum random number generators or QRNG, and quantum key over the internet. D-QKD interfaces have been developed to support the photonic QKD standard ETSI-014. This makes any systems with ETSI QKD standards compatible with D-QKD. D-QKD offers point-to-point quantum entropy and quantum key distributions as well as point-to-multi-points quantum key synchronizations with speeds 1000x faster than photonic QKD. This paper reports benchmark performance tests and randomness quality tests for pure quantum entropy generated by a QRNG and expanded entropy using the QPP protocol. The work has been funded by the PlanQK1project and deployed within the OpenQKD2testbed Berlin, operated by Deutsche Telekom.
Quantum logic gates represent certain quantum operations to perform quantum computations.Of those quantum gates, there is a category of classical behavior gates called quantum permutation gates.As a quantum algorithm, quantum permutation pad or QPP consists of multiple quantum permutation gates to be implemented both in a quantum computing system as a quantum circuit operating on n-qubits' states for transformations and in a classical computing system represented by a pad of n-bit permutation matrices.Since first time proposed in 2020, QPP has been recently applied to create a quantum safe lightweight block cipher by replacing SubBytes and AddRoundKey with QPP in AES called AES-QPP.In AES-QPP, QPP consists of 16 selected 8-bit permutation matrices based on the shared classical key materials.For quantum safe, the key length can be any size from 256 bits to 4 KB.That means, this QPP holds up to 4 KB of Shannon information entropy.Its code size is less than 2 KB with 4 KB of RAM memory.In this paper, we propose to apply QPP for a streaming cipher and carry out its encryption performance and the randomness analysis of this streaming cipher.The proposed QPP streaming cipher demonstrates not only good randomness in its ciphertexts but also huge performance improvement: 13x faster than AES-256, with an overall runtime space (6.8 KB).
Abstract We present an implementation of Kuang and Bettenburg’s Quantum Permutation Pad (QPP) used to encrypt superposition states. The project was conducted on currently available IBM quantum systems using the Qiskit development kit. This work extends previously reported implementation of QPP used to encrypt basis states and demonstrates that application of the QPP scheme is not limited to the encryption of basis states. For this implementation, a pad of 56 2-qubit Permutation matrices was used, providing 256 bits of entropy for the QPP algorithm. An image of a cat was used as the plaintext for this experiment. The plaintext was randomized using a classical XOR function prior to the state preparation procedure. To create corresponding superposition states, we applied a novel operator defined in this paper. These superposition states were then encrypted using QPP, with 2-qubit Permutation Operators, producing superposition ciphertext states. Due to the lack of a quantum channel, we omitted the transmission and executed the decryption procedure on the same IBM quantum system. If a quantum channel existed, the superposition ciphertext states could be transmitted as qubits, and be directly decrypted on a different quantum system. We provide a brief discussion of the security, although the focus of the paper remains on the implementation. Previously we have demonstrated QPP operating in both classical and quantum computers, offering an interesting opportunity to bridge the security gap between classical and quantum systems. This work broadens the applicability of QPP for the encryption of basis states as well as superposition states. We believe that quantum encryption schemes that are not limited to basis states will be integral to a secure quantum internet, to reduce vulnerabilities introduced by using two separate algorithms for secure communication between a quantum and a classical computer.