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This paper conducts a comprehensive benchmarking analysis of the performance of two innovative cryptographic schemes: Homomorphic Polynomial Public Key (HPPK)-Key Encapsulation Mechanism (KEM) and Digital Signature (DS), recently proposed by Kuang et al. These schemes represent a departure from traditional cryptographic paradigms, with HPPK leveraging the security of homomorphic symmetric encryption across two hidden rings without reliance on NP-hard problems. HPPK can be viewed as a specialized variant of Multivariate Public Key Cryptography (MPKC), intricately associated with two vector spaces: the polynomial vector space for the secret exchange and the multivariate vector space for randomized encapsulation. The unique integration of asymmetric, symmetric, and homomorphic cryptography within HPPK necessitates a careful examination of its performance metrics. This study focuses on the thorough benchmarking of HPPK KEM and DS across key cryptographic operations, encompassing key generation, encapsulation, decapsulation, signing, and verification. The results highlight the exceptional efficiency of HPPK, characterized by compact key sizes, cipher sizes, and signature sizes. The use of symmetric encryption in HPPK enhances its overall performance. Key findings underscore the outstanding performance of HPPK KEM and DS across various security levels, emphasizing their superiority in crucial cryptographic operations. This research positions HPPK as a promising and competitive solution for post-quantum cryptographic applications in a wide range of applications, including blockchain, digital currency, and Internet of Things (IoT) devices., Comment: 17 pages

In their 2022 study, Kuang et al. introduced Multivariable Polynomial Public Key (MPPK) cryptography, leveraging the inversion relationship between multiplication and division for quantum-safe public key systems. They extended MPPK into Homomorphic Polynomial Public Key (HPPK), employing homomorphic encryption for large hidden ring operations. Originally designed for key encapsulation (KEM), HPPK's security relies on homomorphic encryption of public polynomials. This paper expands HPPK KEM to a digital signature scheme, facing challenges due to the distinct nature of verification compared to decryption. To adapt HPPK KEM to digital signatures, the authors introduce an extension of the Barrett reduction algorithm, transforming modular multiplications into divisions in the verification equation over a prime field. The extended algorithm non-linearly embeds the signature into public polynomial coefficients, addressing vulnerabilities in earlier MPPK DS schemes. Security analysis demonstrates exponential complexity for private key recovery and forged signature attacks, considering ring bit length twice that of the prime field size., Comment: 16 pages, 1 figure

This paper proposes a new homomorphic functional encryption using modular multiplications over a hidden ring. Unlike traditional homomorphic encryption where users can only passively perform ciphertext addition or multiplication, the homomorphic functional encryption retains homomorphic addition and scalar multiplication properties, but also allows for the user's inputs through polynomial variables. The proposed homomorphic encryption can be applied to any polynomials over a finite field, with their coefficients considered as their privacy. We denote the polynomials before homomorphic encryption as plain polynomials and after homomorphic encryption as cipher polynomials. A cipher polynomial can be evaluated with variables from the finite field, GF(p), by calculating the monomials of variables modulo a prime p. These properties allow functional homomorphic encryption to be used for public key encryption of certain asymmetric cryptosystems to hide the structure of its central map construction. We propose a new variant of MPKC with homomorphic encryption of its public key. We propose to use a single plaintext vector and a noise vector of multiple variables to be associated with the central map, in place of the secret plaintext vector to be encrypted in MPKC. We call this variant of encrypted MPKC, a Homomorphic Polynomial Public Key algorithm or HPPK algorithm. The HPPK algorithm holds the property of indistinguishability under the chosen-plaintext attacks or IND-CPA. The overall classical complexity to crack the HPPK algorithm is exponential in the size of the prime field GF(p). We briefly report on benchmarking performance results using the SUPERCOP toolkit. Benchmarking results demonstrate that HPPK offers rather fast performance, which is comparable and in some cases outperforms the NIST PQC finalists for key generation, encryption, and decryption., Comment: 21 pages, 1 figure

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. To create corresponding superposition states, we applied a novel operator defined in this paper. These superposition states were then encrypted using QPP, 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., Comment: 28 pages, 18 figures, 2 tables, in revision for EPJ

Quantum computing has entered fast development track since Shor's algorithm was proposed in 1994. Multi-cloud services of quantum computing farms are currently available. One of which, IBM quantum computing, presented a road map showing their Kookaburra system with over 4158 qubits will be available in 2025. For the standardization of Post-Quantum Cryptography or PQC, the National Institute of Standards and Technology or NIST recently announced the first candidates for standardization with one algorithm for key encapsulation mechanism (KEM), Kyber, and three algorithms for digital signatures. NIST has also issued a new call for quantum-safe digital signature algorithms due June 1, 2023. This timeline shows that FIPS-certified quantum-safe TLS protocol would take a predictably long time. However, "steal now, crack later" tactic requires protecting data against future quantum threat actors today. NIST recommended the use of a hybrid mode of TLS 1.3 with its extensions to support PQC. The hybrid mode works for certain cases but FIPS certification for the hybridized cryptomodule might still be required. This paper proposes to take a nested mode to enable TLS 1.3 protocol with quantum-safe data, which can be made available today and is FIPS compliant. We discussed the performance impacts of the handshaking phase of the nested TLS 1.3 with PQC and the symmetric encryption phase. The major impact on performance using the nested mode is in the data symmetric encryption with AES. To overcome this performance reduction, we suggest using quantum encryption with a quantum permutation pad for the data encryption with a minor performance reduction of less than 10 percent., Comment: 6 pages, 3 figures, to be submitted for a conference