# Quantum Private Comparison Based on Four-Particle Cluster State
## 1 Introduction  
Classical traditional cryptography, exemplified by RSA and ElGamal, relies on the computational difficulty of specific problems to encrypt information and safeguard private data. However, these classical methods encounter significant security challenges with the emergence of quantum computers. Notably, [PERSON]'s algorithm [1] and [PERSON]'s algorithm [2] pose potential threats to schemes predicated on classical cryptographic techniques, thereby jeopardizing the confidentiality of private information. Quantum mechanics offers distinctive characteristics, including superposition and entanglement, that can be utilized to improve security. In response, researchers have developed quantum cryptographic protocols that integrate classical cryptography with quantum mechanics to bolster security. Numerous quantum cryptographic protocols have been suggested, including quantum key distribution [3; 4; 5], quantum key agreement [6; 7], quantum secure direct communication [8; 9; 10], and quantum private set intersection [11; 12], each tailored to facilitate distinct cryptographic tasks. The private comparison protocol, which is derived from the millionaires' problem, was first proposed by [PERSON] [13]. The objective of the millionaires' problem is to ascertain which of two millionaires possesses greater wealth without revealing their actual financial amounts. [PERSON] et al. [14] extended this problem to focus on the specific issue of determining whether two millionaires possess equal wealth, a scenario known as the socialist millionaires' problem. Since then, solutions to the socialist millionaires' problem have garnered significant scholarly attention. In the context of designing a private comparison protocol, [PERSON] [15] emphasized that securely evaluating a two-party computational functionbetween two parties in a two-party context is fundamentally infeasible. As a result, incorporating a semi-trusted party (STP) becomes essential for enabling the comparison of secrets among users in the development of private comparison protocols. However, these protocols' security remains at risk from quantum-based attacks, as they depend on the computational difficulty associated with factoring large numbers. Therefore, implementing suitable safeguards is crucial to maintaining their security. Unlike classical private comparison protocols, QPC protocols leverage qubits or quantum states as resources instead of classical bits, capitalizing on the unique properties of quantum mechanics rather than depending on computational complexity. This paradigm achieves quantum-based security while ensuring participant privacy. The first QPC protocol was proposed by [PERSON] and [PERSON] [16], employing entangled EPR pairs and unitary operations to facilitate private comparison, while ensuring security through the use of decoy photons and hash functions. Subsequently, [PERSON] et al. [17] introduced an efficient QPC protocol that harnesses the properties of GHZ states, enhancing operational efficiency by partitioning secrets into multiple groups. Since that time, numerous QPC protocols have been developed, employing a range of quantum states as resources. Notably, easily implementable quantum states, such as single photons [18; 19; 20; 21; 22; 23], Bell states [24; 25; 26; 27; 28], and GHZ states [29; 30; 31], have been employed as quantum information carriers for private comparison. Furthermore, some researchers have utilized multiple-qubit entangled states [32; 33], multiple-qubit cluster states [34; 35; 36; 37], and d-dimensional quantum states [38; 39; 40; 41] to facilitate private comparison. In 2020, [PERSON] et al. [42] introduced a QPC protocol based on Bell states, employing quantum gates rather than classical exclusive-OR operations to facilitate private calculations of secrets. This approach enhances security by circumventing traditional computational methods. In 2021, [PERSON] et al.