# Quantum Private Comparison Based on Four-Particle Cluster State
## 1 Introduction
[43] introduced an innovative protocol that utilizes entanglement swapping among three Bell states, allowing for the comparison of three classical bits in each round and attaining a qubit efficiency of 50%. In 2022, [PERSON] et al. [44] utilized rotation encryption and swap tests to ascertain the equality of qubits, rather than comparing numerical values directly. [PERSON] et al. [45] implemented an eight-qubit entangled-state QPC protocol, ensuring input security through quantum key distribution (QKD) protocols that generate secret keys for encrypting confidential information; however, the implementation of eight-qubit states presents greater challenges compared to single photons, Bell states, and GHZ states. In 2023, [PERSON] et al. [46] developed a QPC protocol utilizing 4D GHZ-like states to compare the secrets of two participants with limited quantum capabilities. [PERSON] et al. [47] presented a QPC protocol that utilizes n-particle GHZ states, where unitary operations encode secret information, decoy photons secure the quantum communication channel against external attacks, and hash functions mitigate threats from the semi-trusted party (STP). In 2024, [PERSON] and [PERSON] [48] introduced a QPC protocol using single photons, but preparing multiple sequences of single photons results in reduced qubit efficiency. Additionally, [PERSON] et al. [49] employed Bell states as quantum resources in their QPC protocol, which incorporates rotation operations for classical computations to address the socialist millionaires' problem. Based on the preceding analysis, it is evident that the majority of quantum private comparison (QPC) protocols predominantly focus on comparing the secrets of two users, thereby limiting scalability. To address this issue, we leverage four-particle cluster states, which are comparatively easier to generate and manipulate than larger entangled states. By combining these states with local operations, we propose an efficient QPC protocol, which facilitates the assessment of equality relationships between private information from two groups of users, each consisting of two participants, within a single execution. Each user's input is encoded into the received quantum sequence using bit flip or phase shift operators, which are then transmitted back to the STP. This party determines the equality of the inputs within the same group without gaining knowledge of the individual values and subsequently relays the results back to the users. Our approach improves scalability compared to existing QPC protocols by enabling the simultaneous comparison of private information from two groups. Additionally, our approach achieves a qubit efficiency of 50% by enabling the distributed transmission of the quantum sequence and the practicality of our protocol is further augmented through the utilization of local operations on four-particle cluster states. The subsequent sections are structured as follows: Section 2 offers essential background information. Section 3 details the specific steps of our proposal. Section 4 addresses the security analysis, while Section 5 presents an analysis of efficiency and comparisons. Lastly, Section 6 concludes with a summary of this work.