Technical Papers

MASQ^™

Homomorphic Polynomial Public Key Cryptography for Quantum-secure Digital Signature

By: R. Kuang, M. Perepechaenko

December 2023

DOI: https://doi.org/10.48550/arXiv.2311.08967

… Read More This paper proposes a novel quantum encryption in the coherent optical domain utilizing a displacement operator and implementing with IQ-MZM optical modules, called Quantum Encryption in Phase Space or QEPS. The communication peers share a secret used to seed cryptographic pseudo random number generators to produce a synchronized random number at both the transmitter and receiver. The synchronized random numbers are used to establish displacement operators to encrypt the coherent states at the transmission and decrypt the cipher coherent states at the receiver. Therefore, malicious parties tapping along the fibre line would not extract the message in transit from optical domain due to a high Bit Error Rate or BER. The optimal displacement operator is split into a standard 16-QAM and a random phase shift operator to enhance the transmission security. We analysis the transmission security with the wiretap channel model for semantic security. We have simulated the QEPS encryption and decryption for two data modulation schemes: QPSK and 16-QAM over 80 km for transmission speeds of 56 Gbps for QPSK and 112 Gbps for 16-QAM.

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Homomorphic polynomial public key encapsulation over two hidden rings for quantum- safe key encapsulation

By: R. Kuang, M. Perepechaenko

August 2023

DOI: 10.1007/s11128-023-04064-4

Abstract:
Kuang et al. recently introduced a novel quantum-safe public key scheme, called the multivariate Polynomial Public Key or MPPK. MPPK is based upon the mutual inversion relationship of multiplication and division, with the former used for key pair construction, and the latter used for decryption. For key pair construction, two solvable univariate polynomials are each multiplied by a base multivariate polynomial used for the purpose of noise injection.

… Read More The constant term and highest order term of the produced product polynomials with respect to the message variable are set aside and used to create two noise functions, concealed using a hidden ring. The remaining parts of the product polynomials and two noise functions constitute the public key. The operation used to create noise functions is partially homomorphic. In this paper, we propose to extend the key construction to use this partially homomorphic operator and two hidden rings to hide the public key product polynomials, one for each polynomial. In other words, we propose to encrypt the product polynomials in their entirety with a pair of hidden rings using the partially homomorphic operator. Encrypting the public key this way complicates possible attacks on the public key and forces the adversary to guess the pair of hidden rings. We name this new construction Homomorphic Polynomial Public Key over Two Hidden Rings or HPPK-THR. HPPK-THR demonstrates the IND-CPA property with uninterpretable security in secret recovery attacks, due to the modular Diophantine Equation Problem. In our brief benchmark performance, HPPK-THR outperforms MPPK KEM and NIST Round 3 finalists.

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Benchmark Performance of the Multivariate Polynomial Public Key Encapsulation Mechanism

By: R. Kuang, M. Perepechaenko, M. Barbeau, Toth

May 2023

DOI: 10.1007/978-3-031-31108-6_18

Abstract:
This paper presents the results of benchmarking the quantum-safe Multivariate Public Key Cryptosystem (MPPK) key encapsulation mechanism for quadratic solvable univariate polynomials. We used a benchmarking tool containing implementations of the four NIST Post-Quantum Cryptography (PQC) finalists: Kyber, McEliece, NTRU, and Saber. The benchmark demonstrates that the performance of MPPK is comparable with that of the four PQC algorithms, offering relatively fast key generation and small key sizes. Key encapsulation and decapsulation performance are comparable with the PQC schemes, with room for improvement.

Optimization of the multivariate polynomial public key for quantum safe digital signature

By: R. Kuang, M. Perepechaenko

April 2023

DOI: 10.1038/s41598-023-32461-3

Abstract:
Kuang, Perepechaenko, and Barbeau recently proposed a novel quantum-safe digital signature algorithm called Multivariate Polynomial Public Key or MPPK/DS. The key construction originated with two univariate polynomials and one base multivariate polynomial defined over a ring. The variable in the univariate polynomials represents a plain message. All but one variable in the multivariate polynomial refer to noise used to obscure private information.

… Read More These polynomials are then used to produce two multivariate product polynomials, while excluding the constant term and highest order term with respect to the message variable. The excluded terms are used to create two noise functions. Then four produced polynomials, masked with two randomly chosen even numbers over the ring, form the Public Key. The two univariate polynomials and two randomly chosen numbers, behaving as an encryption key to obscure public polynomials, form the Private Key. The verification equation is derived from multiplying all of the original polynomials together. MPPK/DS uses a special safe prime to prevent private key recovery attacks over the ring, forcing adversaries to solve for private values over a sub-prime field and lift the solutions to the original ring. Lifting entire solutions from the sub-prime field to the ring is designed to be difficult based on security requirements. This paper intends to optimize MPPK/DS to reduce the signature size by a fifth. We added extra two private elements to further increase the complexity of the private key recovery attack. However, we show in our newly identified optimal attack that these extra private elements do not have any effect on the complexity of the private recovery attack due to the intrinsic feature of MPPK/DS. The optimal key-recovery attack reduces to a Modular Diophantine Equation Problem or MDEP with more than one unknown variables for a single equation. MDEP is a well-known NP-complete problem, producing a set with many equally-likely solutions, so the attacker would have to make a decision to choose the correct solution from the entire list. By purposely choosing the field size and the order of the univariate polynomials, we can achieve the desired security level. We also identified a new deterministic attack on the coefficients of two univariate private polynomials using intercepted signatures, which forms a overdetermined set of homogeneous cubic equations. To the best of our knowledge, the solution to such a problem is to brute force search all unknown variables and verify the obtained solutions. With those optimizations, MPPK/DS can offer extra security of 384 bit entropy at 128 bit field with a public key size being 256 bytes and signature size 128 or 256 bytes using SHA256 or SHA512 as the hash function respectively.

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Benchmark Performance of a New Quantum-Safe Multivariate Polynomial Digital Signature Algorithm

By: R. Kuang, M. Perepechaenko, R.Toth, M. Barbeau

November 2022

DOI: 10.1109/QCE53715.2022.00067

Abstract:
Kuang et al. introduced the new quantum-safe algorithm Multivariate Polynomial Public Key Digital Signature (MPPK DS). To create a signature, the MPPK DS scheme’s private key consists of univariate polynomials used as exponents of a secret randomly generated base. For signature verification, the verifier leverages public key multivariate polynomials and a modular arithmetic property.

… Read More The verification procedure is probabilistic. The verifier uses noise variables and evaluates the public key polynomials. For a genuine signature, the verification procedure is successful for any evaluation of the public key polynomials. In this paper, we report the results of benchmarking MPPK DS on a 16-core Intel®Core™i7-10700 CPU system at 2.90 GHz using the SUPERCOP toolkit. SUPERCOP has been widely used to analyze the performance of post-quantum public-key encryption and key-establishment algorithms. We provide a side-by-side comparison of the NIST PQC third-round digital signature schemes with MPPK DS. With respect to the PQC schemes, the MPPK DS cryptosystem achieves small size public keys, private keys, and signatures. Moreover, compared with the NIST PQC digital signature algorithms, the performance of the MPPK DS algorithm is outstanding with fast procedures for key generation, signing, and verifying.

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A new post-quantum multivariate polynomial public key encapsulation algorithm

By: R. Kuang, M. Perepechaenko, M. Barbeau

October 2022

DOI: 10.1007/s11128-022-03712-5

Abstract:
We propose a new quantum-safe cryptosystem called multivariate polynomial public key (MPPK). Its security stems from the hardness of finding integer solutions to multivariate equations over a prime field GF(p). Indeed, for a large prime p, solving modular Diophantine equations is an NP-complete problem.

… Read More MPPK introduces a novel way of key pair generation that involves multiplying a base n-degree multiplicand multivariate polynomial with respect to a message variable and two univariate multiplier polynomials, solvable by radicals over GF(p). The coefficients of the two resulting polynomial products are used to construct the public key, except for the coefficients of the constant and highest degree terms with respect to the message variable. The base multivariate polynomial’s constant and highest degree terms are used to form two noise functions, as parts of the public key, through multiplications with random variables. The private key consists of the two multiplier polynomials and the two random noise constants. MPPK encryption performs multivariate polynomial evaluations with a randomly chosen secret as the message variable and multiple noise values for other variables. The ciphertext tuple is created by calculating the values of two product multivariate polynomials and two noise functions. MPPK decryption eliminates the base multivariate polynomial by dividing by the two-product multivariate polynomial values and then extracting the secret from the resulting univariate polynomial with a radical. For adversarial extraction of the private key from the public key alone, the best complexity is exponential with respect to the bit length of the prime finite field. The same holds for the adversarial extraction of the plaintext from the ciphertext.

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A new quantum-safe multivariate polynomial public key digital signature algorithm

By: R. Kuang, M. Perepechaenko, M. Barbeau

August 2022

DOI: 10.1038/s41598-022-15843-x.

Abstract:
We propose a new quantum-safe digital signature algorithm called Multivariate Polynomial Public Key Digital Signature (MPPK/DS). The core of the algorithm is based on the modular arithmetic property that for a given element g, greater than equal to two, in a prime Galois field GF(p) and two multivariate polynomials P and Q, if P is equal to Q modulo p-1, then g to the power of P is equal to g to the power of Q modulo p.

… Read More MPPK/DS is designed to withstand the key-only, chosen-message, and known-message attacks. Most importantly, making secret the element g disfavors quantum computers’ capability to solve the discrete logarithm problem. The security of the MPPK/DS algorithm stems from choosing a prime p associated with the field GF(p), such that p is a sum of a product of an odd prime number q multiplied with a power x of two and one. Given such a choice of a prime, choosing even coefficients of the publicly available polynomials makes it hard to find any private information modulo p-1. Moreover, it makes it exponentially hard to lift the solutions found modulo q to the ring of integers modulo p-1 by properly arranging x and q. However, finding private information modulo the components q and power x of two is an NP-hard problem since it involves solving multivariate equations over the chosen finite field. The time complexity of searching a private key from a public key or signatures is exponential over GF(p). The time complexity of perpetrating a spoofing attack is also exponential for a field GF(p). MPPK/DS can achieve all three NIST security levels with optimized choices of multivariate polynomials and the generalized safe prime p.

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Digital Signature Performance of a New Quantum Safe Multivariate Polynomial Public Key Algorithm

By: R. Kuang, M. Perepechaenko

April 2022

DOI: 10.1109/ICCCS55155.2022.9846785

Abstract:
We discuss the performance of a new quantumsafe multivariate digital signature scheme proposed recently, called the Multivariate Polynomial Public Key Digital Signature (MPPK DS) scheme. Leveraging MPPK KEM or key exchange mechanism, the MPPK DS scheme is established using modular exponentiation with a randomly chosen secret base from a prime field. … Read More The security of the MPPK DS algorithm largely benefits from a generalized safe prime associated with the said field and the Euler totient function. We can achieve NIST security levels I, III, and V over a 64-bit prime field, with relatively small public key sizes of 128 bytes, 192 bytes, and 256 bytes for security levels I, III, and V, respectively. The signature sizes are 80 bytes for level I, 120 bytes for level III, and 160 bytes for level V. The MPPK DS scheme offers probabilistic procedures for signing and verification. That is, for each given signing message, a signer can randomly pick a base integer to be used for modular exponentiation with a private key, and a verifier can verify the signature with the digital message, based on the verification relationship, using any randomly selected noise variables. The verification process can be repeated as many times as the verifier wishes for different noise values, however, for a true honest signature, the verification will always pass. This probabilistic feature largely restricts an adversary to perform spoofing attacks. In this paper, we conduct some performance analyses by implementing MPPK DS in Java. We compare its performance with benchmark performances of NIST PQC Round 3 finalists: Rainbow, Dilithium, and Falcon. Overall, the MPPK DS scheme demonstrates equivalent or better performance, and much smaller public key, as well as signature sizes, compared to the three NIST PQC Round 3 finalists.

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A Deterministic Polynomial Public Key Algorithm over a Prime Galois Field GF(p)

By: R. Kuang

January 2022

DOI: 10.1109/ACCC54619.2021.00020.

Abstract: The ancient Vieta’s formulas reveal the relationships between coefficients of an nth-degree polynomial and its roots. It is surprisingly found that there exists a hidden secret for a potential public key exchange: decoupling the product of all roots or constant term from summations of root products or coefficients of a polynomial to establish a keypair. The proposed deterministic polynomial public key algorithm or DPPK … Read More is built on the fact that a polynomial cannot be factorized without its constant term. DPPK allows the keypair generator to combine a base polynomial, eliminable during the decryption, with two solvable polynomials and creates two entangled polynomials. Two coefficient vectors of the entangled polynomials form a public key, and their constant terms, together with the two solvable polynomials, form the private key. By only publishing coefficients of polynomials without their constant terms, we greatly restrict polynomial factoring techniques for the private key extraction. We demonstrate that the time complexity, in terms of field operations, of the private key extraction from the known public key is a super-exponential difficulty O(p2) for classical attacks and an exponential difficulty O(p) for quantum attacks, respectively, in comparison with the low sub-exponential complexity for the PQC algorithms. The best-known deterministic complexity of the polynomial factoring problem for the secret key extraction from intercepted ciphertexts is O(npl/2) for classical attacks and O(pl/2) for quantum attacks, respectively, at the same complexity level as Grover’s search algorithm. Performance comparisons with the PQC finalists for keypair generations, encryptions, and decryptions are presented.

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Indistinguishability and Non-deterministic Encryption of the Quantum Safe Multivariate Polynomial Public Key Cryptographic System

By: R. Kuang and M. Barbeau

October 2021

DOI: 10.1109/CCECE53047.2021.9569200.

Abstract:
Multivariate Polynomial Public Key (MPPK) is a cryptographic system, over a prime Galois field. A key pair is generated using a multiplier multivariate polynomial and two multiplicand univariate solvable polynomials. They yield two product multivariate polynomials. The first variable is used for carrying the message or secret and others are used as noise sources. … Read More The public key consists of all the coefficients of the product multivariate polynomials, except the two constant coefficients, in terms with coefficients attached to the message variable, and a noise function or a polynomial of only noise variables generated from the constant term of the multiplier multivariate polynomial by multiplying a private random variable R. The private key is made of both univariate solvable multiplicand polynomials and the private R. Encryption takes a secret message and random numbers for noises, adding noise that is automatically cancelled by decryption. Decryption is achieved evaluating a solvable equation. We review security analysis that can be employed to crack MPPK secrets and private keys. Finally, we discuss indistinguishability and non-deterministic encryption, key properties of MPPK.

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Performance Analysis of the Quantum Safe Multivariate Polynomial Public Key Algorithm

By: R. Kuang and M. Barbeau

November 2021

DOI: 10.1109/QCE52317.2021.00052.

Abstract:
The Multivariate Polynomial Public Key (MPPK) algorithm, over a prime Galois field, takes a multiplier multivariate polynomial and two multiplicand univariate solvable polynomials to create two product multivariate polynomials. One of variables is for secret message and all others are for noises. The public key consists of all coefficients of the product multivariate polynomials, … Read More except the two constant terms for the message variable. The private key is made of both multiplicands. Encryption takes a list of random numbers, over the prime Galois field. The first number is the secret to exchange. The other random numbers generate noise automatically cancelled by decryption. The secret is easily extracted from the evaluation of a solvable equation. The level of security provided by MPPK is adaptable. The algorithm can be used in several different ways. In this paper, we review the performance achieved by MPPK for several combinations of polynomial configurations and Galois field sizes. For every combination, we calculated key generation time, encryption time and decryption time. We also compare the effectiveness of MPPK with the performance of all four NIST PQC finalists. For MPPK, the data has been collected from the execution of an implementation in Java. In comparison to the NIST PQC finalists, MPPK key generation, encryption and decryption performance is excellent.

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QEEP^™

Raw QPP-RNG randomness via system jitter across platforms: a NIST SP 800-90B evaluation

By: R. Kuang, Georgia Vrana,Dafu Lou

July 2025

DOI: 10.1038/s41598-025-13135-8.

Abstract:
High-quality randomness is fundamental to the security of modern cryptographic systems. We present QPP-RNG, a true random number generator (TRNG) that harvests entropy from diverse system-level jitters–including CPU pipeline timing divergences, DRAM refresh cycle perturbations, cache miss-driven memory access latencies, and other subtle hardware and operating system-induced fluctuations. QPP-RNG’s core mechanism measures the elapsed time of randomized array sorting operations–where each Fisher-Yates shuffle is infinitesimally perturbed by these microscopic jitters–and amplifies these timing variations into cryptographically strong randomness through a quantum permutation pad (QPP) architecture, all achievable on commodity hardware.

… Read More The raw output of QPP-RNG underwent rigorous evaluation for independent and identically distributed (IID) behavior using the NIST SP 800-90B IID test suite, alongside the comprehensive NIST SP 800-22 and ENT statistical test batteries. Across a range of platforms, including Windows, macOS, and Raspberry Pi, QPP-RNG consistently achieved high IID min-entropy between and bits/byte. It passed all NIST SP 800-90B IID tests with -values significantly above the threshold, confirming that its generated randomness is statistically indistinguishable from ideal IID sources derived directly from system jitter. Cross-platform analyses spanning x86_64 and ARM64 architectures further demonstrate that the extracted jitter fingerprint–and consequently the generated randomness–exhibits remarkable statistical consistency, irrespective of the underlying hardware or operating system. QPP-RNG’s entropy density compares favorably with leading commercial entropy sources. It matches or slightly exceeds the NIST IID-certified min-entropy of ID Quantique’s Quantis QRNG (7.8744 bits/byte) and significantly outperforms both Red Hat’s CPU Time Jitter RNG (7.4528 bits/byte) and Quside’s PCIe One quantum entropy source (6.5136 bits/byte). Even against specialized hardware RNGs like Microchip’s ECC608 (4.0568 bits/byte), QPP-RNG demonstrates superior performance using only general-purpose processors. By effectively transforming otherwise discarded system noise into a reliable and high-quality entropy stream, QPP-RNG establishes a novel paradigm for embedded security, providing a robust entropy source on general-purpose devices without specialized hardware. This makes it especially well-suited for resource-constrained Internet of Things (IoT) and edge computing applications where strong entropy sources are paramount.

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Quantum permutation pad for universal quantum-safe cryptography

By: R. Kuang, M. Barbeau

June 2022

DOI: 10.1007/s11128-022-03557-y.

Abstract:
Classical cryptographic techniques are currently under the growing quantum computing threat. New techniques that quantum computing algorithms cannot break are urgently needed. We present such an encryption method. It builds upon quantum permutation logic gates or quantum permutation pads. It is universal in that it can be equally employed on classical computers, … Read More today’s Internet, and the upcoming quantum Internet. While the cryptographic technique is formulated in a quantum computing framework, it does not rely on physical properties uniquely present at the quantum level, such as no-cloning or entanglement of data. It achieves with today’s technology a level of security comparable to what will be possible to attain with tomorrow’s quantum technology. The mathematics behind the cryptographic technique, quantum representations of a symmetric group over a computational basis, is surprisingly simple. However, the challenge faced by an adversary wishing to break the code is intractable and uninterpretable, a property of Shannon’s perfect secrecy. We believe that the cryptographic technique presented in this article can be used in several different ways and modes. It can be integrated into numerous current Internet protocols, or the Internet of Things, making them quantum safe. In addition, it can be used to transition to the upcoming Internet quantum technology smoothly.

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Quantum permutation pad for universal quantum-safe cryptography

By: R. Kuang, M. Barbeau

June 2022

DOI: 10.1007/s11128-022-03557-y.

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Quantum Encrypted Communication between Two IBMQ Systems Using Quantum Permutation Pad

By: R. Kuang, D. Lou, M. Perepechaenko

July 2022

DOI: 10.1109/ICCCAS55266.2022.9824836.

Abstract:
We demonstrate an early functional implementation of the Kuang and Barbeau’s Quantum Permutation Pad (QPP) algorithm on the IBMQ physical quantum computers using the Qiskit development kit. Our implementation of the quantum encryption QPP algorithm uses 2-qubit permutation operators created from the compositions of a few CNOT and NOT gates.

… Read More With inability to physically transfer qubits between two IBMQ machines we acted as two separate IBMQ computers and described how ideally two quantum systems can securely communicate using QPP. Since the physical qubits are still noisy, we use a simple error correction technique by choosing the correct state with the highest probability. Our implementation can be extended to a hybrid system that consist of a quantum computer communicating with a classical computer securely using QPP. This work can be considered as a toy example of the fully secure implementation of quantum encryption using QPP. Nevertheless, it is a promising first step towards secure quantum communication.

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Quantum secure lightweight cryptography with quantum permutation pad

By: R. Kuang, D. Lou, A. He, and A. Conlon

August 2021

DOI: 10.25046/aj060445.

Abstract: 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 … Read More 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).

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Shannon Perfect Secrecy in a Discrete Hilbert Space

By: R. Kuang and N. Bettenburg

November 2020

DOI: 10.1109/QCE49297.2020.00039.

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Quantum Safe Lightweight Cryptography with Quantum Permutation Pad

By: R. Kuang, D. Lou, A. He, and A. Conlon

June 2021

DOI: 10.1109/ICCCS52626.2021.9449247.

Abstract:
Quantum permutation pad or QPP was first proposed by Kuang and Bettenburg in 2020 [15]. QPP is a generic quantum algorithm consisting of multiple n-qubits quantum permutation gates. As a quantum algorithm, QPP can be implemented both in a quantum computing system as a quantum circuit operating on n-qubits’ state for transformation and in a classical … Read Morecomputing system represented by a pad of n-bit permutation matrices. QPP has two unique characteristics: huge Shannon information entropy and non-commutativity between permutation matrices or the generalized uncertainty principal. Permutation transformation is bijective mapping between input information space and output ciphertext space. That means, QPP has the property of Shannon perfect secrecy with reusability due to the uncertainty relationship. QPP is the generalization of One-Time-Pad or OTP over Hilbert space and OTP is the simplification of QPP over a Galois field. Based on those, this paper explores a variant of AES for a quantum safe lightweight cryptography by incorporating AES ShiftRows and MixColumns with QPP or called AES-QPP. AES-QPP unifies the SubBytes and AddRoundKey with the same QPP of 16 8-bit permutation matrices, essentially SubBytes to be a special 8-bit permutation matrix and AddRoundKey to be 16 8-bit permutation matrices selected from XOR operations. By randomly selecting 16 permutation matrices with a secret key material, AES-QPP could hold a total equivalent 26,944 bits of Shannon entropy. It not only improves the security against differential and linear attacks but also largely reduces the number of rounds to 5 rounds. AES-QPP could be a good candidate for quantum safe lightweight cryptography.

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SEQUR^™

Benchmark Performance of Digital QKD Platform Using Quantum Permutation Pad

By: A. He, M. Redding, M. Geitz, R. Toth, R. Döring, R. Carson, R. Kuang

October 2022

DOI: 10.1109/ACCESS.2022.3212738.

Abstract:
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
… Read More 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 PlanQK1 project and deployed within the OpenQKD2 testbed Berlin, operated by Deutsche Telekom.

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Entropy Transformation and Expansion with Quantum Permutation Pad for 5G Secure Networks

By: R. Kuang, D. Lou, A. He

October 2021

DOI: 10.1109/ICCT52962.2021.9657891.

Abstract:
This paper proposes a quantum safe solution for 5G networks using a quantum permutation pad (or QPP) algorithm, originated from quantum computing logic gates or quantum permutation gates. All permutation gates form a unique permutation space, just like a classical key space. An n-bit permutation space consists of the entire 2 n ! permutation gates, or permutation matrices … Read More over its computational basis. The huge equivalent Shannon entropy of this permutation space would be a nice entropy source for information security. Kuang and Bettenburg in 2020 first proposed QPP and proved it to be the Shannon type of perfect secrecy. Here, we extend its capability of entropy transformations for distribution over the Internet to entropy expansions for 5G networks. We analyze the randomnesses following transformations and expansions with QPP, using industry randomness testing suites. Testing results confirm that QPP can maintain the original randomness of QRNG random numbers for transformations and expansions. Leveraging its strong diffusion capability, QPP may also improve the byte-level randomness of input random numbers.

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Generating Just-in-Time Shared Keys (JIT-SK) for TLS 1.3 Zero RoundTrip Time (0-RTT)

By: E. AbdAllah, R. Kuang, and C. Huang

December 2021

DOI: 10.18178/ijmlc.2022.12.3.1086.

Abstract:
The main goal of Transport Layer Security (TLS) protocol is to provide a secure communication channel between communicating pairs. A new version of the protocol, TLS 1.3, is introduced to improve security and performance for customers. One of the major advantages of TLS 1.3 over earlier versions is that it introduces Zero RoundTrip Time (0-RTT) feature, that saves a round trip at connection setup stage. 0-RTT data … Read Moresecurity properties are weaker than other kinds of TLS data because the data is not forward secret and it is vulnerable to replay attacks. Existing solutions such as single-use tickets, client hello recording, and freshness checks provide inefficient solutions for 0-RTT problems. In this paper, we propose an efficient technique to utilize 0-RTT feature with forward secrecy and prevent replay attacks. Our technique uses a synchronized pseudorandom number generator (PRNG) that depends on initial shared secret between communicating pairs. The initial secret can be shared using TLS 1.3 three basic key exchange modes. In our technique, the PRNG also uses session shared information such as session ID to dynamically provide Just-in-Time Shared Keys (JIT-SK) for 0-RTT. Client and server sides change the keys in each session and hence securely and efficiently use the 0-RTT. We implement a proof of concept for our technique using our private PRNG, named Quantum Entropy Expansion and Propagation (QEEP), and WolfSSL implementation for TLS 1.3 and show the differences using our solution.

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Pseudo Quantum Random Number Generator with Quantum Permutation Pad

By: R. Kuang, D. Lou, A. He, C. McKenzie and M. Redding

June 2021

DOI: 10.1109/QCE52317.2021.00053.

Abstract:
Cryptographic random number generation is critical for any quantum-safe encryption. Based on the natural uncertainty of some quantum processes, a variety of quantum random number generators, or QRNGs, have been created with physical quantum processes. These typically generate random numbers with good unpredictable randomness. Of course, physical … Read MoreQRNGs are costic and require physical integrations with computing systems. This paper proposes a pseudo quantum random number generator with a quantum algorithm called a quantum permutation pad, or QPP, leveraging the high entropy of quantum permutation space for its bijective transformation. Unlike Boolean algebra, where the size of information space is 2ⁿ for an n-bit system, an n-bit quantum permutation space consists of 2ⁿ! quantum permutation matrices, representing all quantum permutation gates over an n-bit computational basis. This permutation space holds an equivalent Shannon information entropy of log₂(2ⁿ!). A QPP can be used to create a pseudo-QRNG or pQRNG capable of integration with any classical computing system, or directly with any application, for good-quality deterministic random number generation. Using a QPP pad with 64 8-bit permuation matrices, a pQRNG holds 107,776 bits of entropy for pseudo-random number generation, compared with 4,096 bits of entropy in Linux /dev/random. It can be used as a deterministic PRNG or as an entropy booster for other PRNGs. It can also be used as a whitening algorithm for any hardware random number generator, including QRNGs, without discarding physical bias bits.

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QXD

Experimental demonstration of quantum encryption in phase space with displacement operator in coherent optical communications

By: A. Chan, M. Khalil, D. Plant, R. Kuang

July 2024

DOI: https://doi.org/10.1140/epjqt/s40507-024-00260-y

Abstract:
We provide experimental validation of quantum encryption in phase space using displacement operators in coherent states (DOCS) in a conventional coherent optical communication system. The proposed encryption technique is based on displacing the information symbols in the phase space using random phases and amplitudes to achieve encryption randomly and provide security at the physical layer. … Read More We also introduce a dual polarization encryption approach where we use two different and random DOCS to encrypt the X and Y polarizations separately. The experimental results show that only authorized users can decrypt the signal correctly, and any mismatch in the displacement operator coefficients, amplitudes, or phases will lead to a bit error ratio (BER) of approximately 50%. We also compare the performance of the system with and without encryption over 80 km of standard-single mode fiber (SSMF) transmission to assess the added penalty of such encryption. The achieved net bit rates are 224, 448, and 560 Gb/s for QPSK, 16QAM, and 32QAM modulation formats, respectively. The experimental results showcase the efficacy of the DOCS encryption technique in resisting various decryption attempts, demonstrating its effectiveness in ensuring the security and confidentiality of transmitted data in a real-world transmission scenario.

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Quantum Encryption in Phase Space using Displacement Operator for QPSK Data Modulation

By: R. Kuang, Adrian Chan

March 2023

DOI: 10.48550/arXiv.2301.02894.

Abstract: In 2020, Kuang and Bettenburg proposed Quantum Public Key Distribution (QPKE) which utilized the randomized phase shift gate. Since then, it has been implemented both theoretically through simulations and experimentally over existing fiber optical networks. QPKE can be compared to an RSA-type scheme but in the optical analogue domain. Later on, it was renamed Quantum Encryption in Phase Space (QEPS) to emphasize the encryption of coherent states in phase space. However, the phase shift gate used in QEPS is limited to data modulation schemes with phase shift keying such as quadrature phase shift keying (QPSK) as it may leak data information in amplitude if applied to quadrature amplitude modulation (QAM) schemes. … Read More Recently, Kuang and Chan proposed a new version of QEPS known as Quantum Encryption in Phase Space with the displacement gate or QEPS-d, which overcomes the limitation of QEPS with the phase shift gate. This was achieved by introducing a reduced displacement operator that ignores the global phase factor, making the reduced displacement operators commutable, thus aiding the implementation at both transmission and reception. Furthermore, any arbitrary displacement operator can be decoupled into a standard QAM modulation with a phase shift modulation, making encryption and decryption easier. This paper demonstrates the simulation of QEPS-d encryption for QPSK data modulation to illustrate how QEPS-d functions.

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Quantum encryption in phase space with displacement operators

By: A. Chan, R. Kuang

December 2022

DOI: 10.1140/epjqt/s40507-023-00183-0.

Abstract:
In photonic computing, the quantum systems consist of coherent states and squeezed coherent states. Common quantum gates found in these systems are: phase shift, displacement, and squeezing gates. These gates are all unitary and reversible. Outside of quantum systems, coherent states also plays a significant role in coherent optical communications with speeds of hundreds of gigabits per second. Secure optical communications is generally implemented at the data layer with classical symmetric encryption such as Advanced Standard Encryption or AES. This inevitably allows any wiretapping to capture the transmitted data either in the plaintext mode or in the encrypted ciphertext mode in the optical infrastructure. The recent and rapid developments in Quantum computing further lift up the need for quantum secure communications in the optical infrastructure. … Read More This paper proposes a novel quantum encryption in the coherent optical domain utilizing a displacement operator and implementing with IQ-MZM optical modules, called Quantum Encryption in Phase Space or QEPS. The communication peers share a secret used to seed cryptographic pseudo random number generators to produce a synchronized random number at both the transmitter and receiver. The synchronized random numbers are used to establish displacement operators to encrypt the coherent states at the transmission and decrypt the cipher coherent states at the receiver. Therefore, malicious parties tapping along the fibre line would not extract the message in transit from optical domain due to a high Bit Error Rate or BER. The optimal displacement operator is split into a standard 16-QAM and a random phase shift operator to enhance the transmission security. We analysis the transmission security with the wiretap channel model for semantic security. We have simulated the QEPS encryption and decryption for two data modulation schemes: QPSK and 16-QAM over 80 km for transmission speeds of 56 Gbps for QPSK and 112 Gbps for 16-QAM.

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Security Analysis of a Next Generation TF-QKD for Secure Public Key Distribution with Coherent Detection over Classical Optical Fiber Networks

By: A. Chan, M. Khalil, K. A. Shahriar, L. Chen, D. Plant, R. Kuang

December 2021

DOI: 10.1109/ICCC54389.2021.9674320.

Abstract:
Twin-field quantum key distribution (TF-QKD) has recently attracted attention for its ability to overcome the fundamental limits of secret key rate for point-to-point connectivity without quantum repeaters in QKD. Coherent-based TF-QKD or CTF-QKD, which utilizes coherent states for both transmissions and detections, has allowed systems to be designed for existing optical fiber communication … Read More networks allowing for improved performance compared to conventional QKD. Here, we report a theoretical study of CTF-QKD security from an eavesdropper. Compared to conventional QKD and TF-QKD systems, CTF-QKD system is not limited to using only single photon sources/detectors allowing this system to have comparable performance and range as current optical fiber networks. Using commercially available simulation software, we verify the efficacy by studying three different preventative measures for different modulation formats to prevent an eavesdropper from obtaining any secret key information. Results show that parameters can be limited to prevent an eavesdropper from obtaining any information. The simulation also demonstrates that the photon number-phase uncertainty principle for coherent states starts to play major role of security at 128-QAM modulation. Additional security measures are also described to detect the presence of an eavesdropper and improve the system integrity.

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Security Performance of Public Key Distribution in Coherent Optical Communications Links

By: M. Khalil, A. Chan, K. A. Shahriar, L. R. Chen, D. V. Plant and R. Kuang

July 2021

DOI: 10.1109/ICCCI51764.2021.9486822.

Abstract:
Theoretical uncertainties associated with the measurement of quantum bits have made Quantum Key Distribution (QKD) an attractive research topic for secure communication systems. Recent research in this area has focused on using weak coherent laser pulses as the carrier instead of the traditional approach of using single photon carriers. This opens the possibility to implement QKD infrastructure that is compatible with the existing optical fiber communication network. In this work, we report a theoretical study of a new approach that utilizes the Quantum Public Key Envelope (QPKE) based on the randomized Glauber states in a classical coherent optical communication system to obtain a secure communication. … Read More We verify the efficacy of the system using numerical simulation. We also study the possible approaches for an eavesdropper to test the degree of security enhancement of the proposed system at bitrate of 56 Gb/s. Two eavesdropping techniques are proposed for security analysis. A set of security parameters are used to simulate the variety of eavesdropping schemes.

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Quantum Public Key Distribution using Randomized Glauber States

By: R. Kuang and N. Bettenburg

November 2020

DOI: 10.1109/QCE49297.2020.00032.

Abstract:
State-of-the-art Quantum Key Distribution (QKD) is based on the uncertainty principle of qubits on quantum measurements and is theoretically proven to be unconditionally secure. Over the past three decades, QKD has been explored with single photons as the information carrier. More recently, attention has shifted towards using weak coherent laser pulses … Read Moreas the information carrier. In this paper, we propose a novel quantum key distribution mechanism over a pure optical channel using randomized Glauber states. The proposed mechanism closely resembles a quantum mechanical implementation of the public key envelope idea. The core idea can be described in five steps as follows: 1. A user (Bob) generates a Glauber state as a quantum public key envelope (QPKE) by randomly modulating a secret phase φr, known only to Bob, and transmits it over an optical channel to the other user (Alice). 2. Alice modulates a key phase φk into the QPKE based on a random key and selected modulation scheme and returns it to Bob. 3. For the returning QPKE, Bob derandomizes it with his private key or the phase -φr and then 4. passes it to a coherent receiver to measure the key phase φk. 5. For better security, differential phase-shift keying (DPSK) technique with a reference list is applied to extract keys. For the proposed solution, we explore physical countermeasures to provide path authentication and to avoid man-in-the-middle attacks. Other attack vectors can also be effectively mitigated by leveraging the QPKE, the uncertainty principle and the DPSK modulation technique.

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On the Security of an Optical Layer Encryption Using Coherent-based TF-QKD in Classical Optical Fiber Links

By: A. Chan, M. Khalil, K. A. Shahriar, L. R. Chen, D. V. Plant, R. Kuang

August 2020

DOI: 10.1109/ICCCI55554.2022.9850244.

Abstract:
Twin-field quantum key distribution (TF-QKD) protocols have remained an enticing solution because it is able to overcome many of the limitations that are inherently found in QKD. However, the performance of this technology is still not adequate to be implemented for secure transmission of data. … Read More A variant of the TF-QKD protocol is Coherent-based Two Field QKD (CTF-QKD), which utilizes coherent states for both transmission and detections, is integratable with current infrastructure, and has similar performance to current classical optical communication systems. This proceeding is a continuation of the preliminary security analysis that was previously published in [1–2]. Key vulnerabilities in the current system are identified in simulation and guidelines to prevent an eavesdropper from obtaining data from a tapping attack. Finally, the practical security of the system is discussed and the challenges for an attacker to accomplish real-time decryption of data transmitted.

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Technical Papers

MASQ™

By: R. Kuang, M. Perepechaenko

December 2023

By: R. Kuang, M. Perepechaenko

August 2023

By: R. Kuang, M. Perepechaenko, M. Barbeau, Toth

May 2023

By: R. Kuang, M. Perepechaenko

April 2023

By: R. Kuang, M. Perepechaenko, R.Toth, M. Barbeau

November 2022

By: R. Kuang, M. Perepechaenko, M. Barbeau

October 2022

By: R. Kuang, M. Perepechaenko, M. Barbeau

August 2022

By: R. Kuang, M. Perepechaenko

April 2022

By: R. Kuang

January 2022

By: R. Kuang and M. Barbeau

October 2021

By: R. Kuang and M. Barbeau

November 2021

QEEP™

By: R. Kuang, Georgia Vrana,Dafu Lou

July 2025

By: R. Kuang, M. Barbeau

June 2022

By: R. Kuang, M. Barbeau

June 2022

By: R. Kuang, D. Lou, M. Perepechaenko

July 2022

By: R. Kuang, D. Lou, A. He, and A. Conlon

August 2021

By: R. Kuang and N. Bettenburg

November 2020

By: R. Kuang, D. Lou, A. He, and A. Conlon

June 2021

SEQUR™

By: A. He, M. Redding, M. Geitz, R. Toth, R. Döring, R. Carson, R. Kuang

October 2022

By: R. Kuang, D. Lou, A. He

October 2021

By: E. AbdAllah, R. Kuang, and C. Huang

December 2021

By: R. Kuang, D. Lou, A. He, C. McKenzie and M. Redding

June 2021

QXD

By: A. Chan, M. Khalil, D. Plant, R. Kuang

July 2024

By: R. Kuang, Adrian Chan

March 2023

By: A. Chan, R. Kuang

December 2022

By: A. Chan, M. Khalil, K. A. Shahriar, L. Chen, D. Plant, R. Kuang

December 2021

By: M. Khalil, A. Chan, K. A. Shahriar, L. R. Chen, D. V. Plant and R. Kuang

July 2021

By: R. Kuang and N. Bettenburg

November 2020

By: A. Chan, M. Khalil, K. A. Shahriar, L. R. Chen, D. V. Plant, R. Kuang

August 2020

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