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Frontiers in Physics

A novel semi-quantum private comparison protocol of size relation based on d-dimensional Bell states

Yuke Tong, Tao Chen, Hao Cao

Published Jul 24, 2026• 1 citation
Publisher

Abstract

Many semi-quantum private comparison (SQPC) protocols have been proposed to compare the equality of participants, but they suffer from low qubit efficiency and require classical participants to perform quantum state measurement and preparation operations. We here propose a novel SQPC protocol based on d-dimensional Bell states. With the assistance of a semi-honest third party, classical participants can compare the sizes of their privacies merely through particle reflection and simple unitary operations, but they do not need to prepare and measure quantum states. Furthermore, participants can compare two bits of privacy just once with one qubit. By combining decoy state eavesdropping detection with pre-shared key encryption as dual security strategies, analysis shows that the proposed protocol is secure against internal and external attack in theory. The qubit efficiency of the protocol reaches 66.7 % , which is significantly superior to existing schemes.

DOI
10.3389/fphy.2026.1876070

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Many semi-quantum private comparison (SQPC) protocols have been proposed to compare the equality of participants, but they suffer from low qubit efficiency and require classical participants to perform quantum state measurement and preparation operations. We here propose a novel SQPC protocol based on d-dimensional Bell states. With the assistance of a semi-honest third party, classical participants can compare the sizes of their privacies merely through particle reflection and simple unitary operations, but they do not need to prepare and measure quantum states. Furthermore, participants can compare two bits of privacy just once with one qubit. By combining decoy state eavesdropping detection with pre-shared key encryption as dual security strategies, analysis shows that the proposed protocol is secure against internal and external attack in theory. The qubit efficiency of the protocol reaches 66.7 % , which is significantly superior to existing schemes.
A feasible multi-party quantum private comparison (MQPC) protocol based on d -dimensional Bell states was proposed. In the protocol, all participants can independently encrypt their privacies and send them to a semi-honest quantum third party (TP) through authenticated channels. Then, the TP can determine the size relationship among all participants’ privacies without gaining access to the private information. We verified correctness and effectiveness of the proposed protocol with some examples. In addition, compared with other similar protocols, it is not necessary to perform unitary operation on particles and only single-particle measurement is required. Furthermore, the relatively high qubit efficiency is promised. The security analysis verifies that the proposed protocol can counteract external and internal attacks in theory.
Abstract Semi-quantum private comparison (SQPC) is a quantum cryptographic protocol that allows two classical participants to securely determine whether their private inputs are equal, with the assistance of a semi-honest third party possessing full quantum capabilities. However, most existing SQPC protocols suffer from low qubit efficiency and high implementation costs, primarily due to their reliance on pre-shared keys and complex entangled state preparations. To address these limitations, we propose a novel SQPC protocol based on hybrid quantum states, which integrates Bell states and single-qubit states within a circular transmission architecture. The protocol eliminates the requirement for pre-shared keys and reduces consumption of quantum resources by introducing randomness in the placement of quantum states. Security analysis demonstrates that the proposed protocol is robust against measure–resend attacks, entanglement measurement attacks and internal threats from dishonest participants. Furthermore, performance evaluation shows that the proposed protocol achieves a high qubit efficiency, quantified as η = n 10 n + 1 , where n is the number of bits compared. This efficiency surpasses that of several previously proposed SQPC protocols.
Abstract An efficient semi-quantum private comparison protocol is proposed with d -dimensional Bell states to assist two classical parties in comparing the size relation of their secret information. With the assistance of a third party (TP), Alice and Bob do not need to measure qubits or prepare new qubits, reducing the consumption of quantum resources. The presented protocol has a higher qubit efficiency than other similar protocols, since the legal participants can compare two bits of privacy (i.e., secret information unknown to others) each time with one qubit. Moreover, the proposed scheme offers a greater advantage when comparing larger private numbers, with the correctness of its outputs verified through analytical methods. Simulation experiments further verified the correctness of the proposed protocol, and security analyses demonstrated its robustness against external and internal attacks. Furthermore, the proposed semi-quantum private comparison protocol can be converted into an efficient semi-quantum key distribution protocol.
Abstract In this paper, we propose a novel semiquantum private comparison (SQPC) protocol based on Bell states, which enables one quantum user and one classical user to compare the equality of their private inputs with the help of a semi-honest quantum third party (TP). TP is assumed to be semi-honest in the sense that she may take all possible attacks to steal users’ private inputs except conspiring with anyone. The security analysis validates that our protocol can resist not only the attacks from internal participants but also the attacks from an external eavesdropper. Besides, our protocol only asks TP to perform Bell basis measurements but does not need quantum entanglement swapping; and it releases the classical user from conducting quantum measurements and having a quantum memory. Moreover, our protocol can take advantage over previous SQPC protocols based on Bell states in qubit efficiency. Finally, our protocol can be generalized into its counterpart of the collective-dephasing noise quantum channel.
Abstract Based on Bell states, a new semi-quantum private comparison protocol is proposed that enables two classical users to securely compare the equality of their private information with the aid of a semi-honest third party. Different from the existing semi-quantum private comparison protocols, the two classical participants in the presented protocol do not need to measure and prepare any quantum state, which not only reduces the consumption of quantum devices, but also greatly improves the feasibility of the protocol. Performing different unitary operations on the received particles, classical users can securely compare their secret information. Besides, the devised protocol has higher qubit efficiency than the other similar counterparts, since participants can compare a two-bit privacy each time with one qubit. Meanwhile, after completing the comparison process, all Bell states could be reused since they still retain the corresponding entanglement property, which greatly facilitates the recycle of quantum resources. Security analyses indicate that the designed scheme is secure against external attack and internal attack. Moreover, the operations involved in our scheme are simulated on the IBM Quantum Experience to demonstrate the effectiveness and security of our scheme.
In this paper, we propose a novel semiquantum private comparison (SQPC) protocol of the size relationship based on d-level single-particle states. This protocol can compare the size relationship of different privacy messages from two classical users with the help of a semi-honest third party (TP), who is permitted to misbehave on her own, but cannot be in collusion with anyone else. The correctness analysis shows that this protocol can gain correct comparison results. The security analysis results reveal that this protocol can resist some famous outside attacks and participant attacks. This protocol only adopts d-level single-particle states as the initial quantum resource, only requires the TP to perform d-level single-particle measurements, and makes the accurate comparison results inaccessible to the TP. Compared with the only existing SQPC protocol of size relationship (Quantum Inf. Process. 20, 124 (2021)), the proposed protocol has advantages in experimental preparation of initial quantum resource, TP’s experimental measurements, and TP’s knowledge about the comparison results.