# Two-Party Quantum Private Comparison Protocol Based on Rotational Encryption
## 1 Introduction  
The BB84 protocol [1], introduced by [PERSON] and [PERSON], laid the foundation for quantum key distribution (QKD) by demonstrating the potential for unconditional security through quantum mechanics. Since then, various quantum cryptographic protocols have emerged, including quantum secret sharing (QSS) [2, 3, 4], quantum secure direct communication (QSDC) [5, 6, 7, 8], quantum key agreement (QKA) [9, 10, 11], and quantum private set intersection (QPSI) [12, 13, 14, 15], all aimed at achieving classical cryptographic tasks while providing resistance to quantum attacks. Private comparison stems from the millionaires' problem [16], in which two parties seek to identify who is wealthier without disclosing their exact financial statuses. [PERSON] et al. [17] expanded on this concept to address whether the two parties have equal wealth, establishing it as a fundamental issue in secure multiparty computing. In the context of two-party computation, [PERSON] [18] pointed out constructing a secure equality function is not feasible without additional measures. Thus, a semi-honest third party (TP) is introduced to facilitate this task. The security of private comparison protocols, much like that of traditional cryptographic systems, has historically relied on the computational difficulty of specific mathematical problems. However, the emergence of quantum computation presents substantialchallenges to these classical approaches as quantum algorithms can solve these mathematical problems that were previously considered intractable. In response to these emerging threats, researchers have begun to explore and develop innovative private comparison protocols that incorporate quantum principles, aiming to significantly enhance security and resilience against potential quantum attacks. The objective of QPC is to compare two confidential secrets without disclosing any information beyond the outcome of the comparison. The pioneering QPC protocol, developed by [PERSON] and [PERSON] in 2009 [19], utilized Bell states and decoy states to assess the equality of two private secrets, achieving a qubit efficiency of 25%. Following this, [PERSON] et al. [20] employed GHZ states and single-particle measurements for QPC protocol, improving the qubit efficiency to 33% by grouping classical bits into multiple sets. [PERSON] et al. [21] utilized four-particle x-type states as information carriers and Pauli local unitary operations for QPC protocol, although its efficiency is limited to 25%. [PERSON] and [PERSON] [22] suggested a QPC protocol by leveraging five-qubit entangled states, reaching an efficiency of 40% since one five-qubit state can compare two bits. [PERSON] [23] suggested another QPC protocol utilizing six-particle maximally entangled states, achieving an efficiency of 33%, as the equality comparison of two bits requires one six-particle maximally entangled state. [PERSON] et al. [24] utilized maximally entangled seven-qubit states in their QPC protocol, which has an efficiency of approximately 29% because one seven-particle state can compare two bits. [PERSON] et al. [25] utilized eight-qubit entangled states for proposing their QPC protocol, achieving an efficiency of 25%. Moreover, numerous QPC protocols have been created using different types of entangled states, such as Bell states [26; 27; 28; 29; 30; 31], multi-qubit states [32; 33; 34; 35; 36], and d-level quantum states [37; 38; 39; 40]. Upon reviewing the existing QPC protocols, we observe a significant reliance on multi-qubit entangled states. However, challenges like the complexity of preparing these states and their associated security vulnerabilities limit their practical application. In contrast, single photons offer greater practicality for real-world scenarios compared to protocols relying on multiple-qubit or d-dimensional quantum states. Additionally, many current QPC protocols struggle with lower qubit efficiency, with numerous implementations achieving less than 50% efficiency.