Quantum secure privacy-preserving array equality comparison protocol

Min Hou; Yue Wu; Shibin Zhang; Min Hou; Yue Wu; Shibin Zhang

doi:10.3934/math.2026311

AIMS Mathematics

2026, Volume 11, Issue 3: 7573-7592. doi: 10.3934/math.2026311

Previous Article Next Article

Research article

Quantum secure privacy-preserving array equality comparison protocol

1.
School of Computer Science, Sichuan University Jinjiang College, Meishan 620860, China
2.
Advanced Cryptography and System Security Key Laboratory of Sichuan Province, Chengdu 610225, China
3.
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
4.
College of Artificial Intelligence, Chengdu University of Information Technology, Chengdu 610225, China

Received: 20 January 2026 Revised: 03 March 2026 Accepted: 16 March 2026 Published: 23 March 2026
MSC : 81P65, 81P94

This paper presents a quantum protocol for the secure privacy-preserving equality comparison of private arrays. The design overcomes the scalability limitations of conventional quantum private comparison schemes, which are typically restricted to single integers. In the proposed approach, each array element is encoded into the amplitude of a single-photon state. Participants then encrypt the received states via local rotation operations with privately chosen random angles. A circular transmission mode is employed, whereby the encrypted states are sequentially exchanged among all parties, including a semi-honest third party. The TP performs single-particle measurements to determine the equality of the arrays and broadcasts the result simultaneously to ensure fairness. The protocol's functionality and practicality were validated through quantum circuit simulations on the IBM Qiskit platform. Security analysis confirms its robustness against both external eavesdropping attacks and internal privacy threats from honest-but-curious participants. By utilizing single-photon encoding, rotation-based encryption, and single-particle measurement, the protocol achieves enhanced scalability and practical feasibility for multi-element comparisons, advancing beyond the scope of existing single-integer comparison schemes.
- privacy-preserving equality array comparison,
- quantum private comparison,
- circular transmission mode,
- rotation-based encryption,
- single photon
Citation: Min Hou, Yue Wu, Shibin Zhang. Quantum secure privacy-preserving array equality comparison protocol[J]. AIMS Mathematics, 2026, 11(3): 7573-7592. doi: 10.3934/math.2026311

Related Papers:

Abstract

This paper presents a quantum protocol for the secure privacy-preserving equality comparison of private arrays. The design overcomes the scalability limitations of conventional quantum private comparison schemes, which are typically restricted to single integers. In the proposed approach, each array element is encoded into the amplitude of a single-photon state. Participants then encrypt the received states via local rotation operations with privately chosen random angles. A circular transmission mode is employed, whereby the encrypted states are sequentially exchanged among all parties, including a semi-honest third party. The TP performs single-particle measurements to determine the equality of the arrays and broadcasts the result simultaneously to ensure fairness. The protocol's functionality and practicality were validated through quantum circuit simulations on the IBM Qiskit platform. Security analysis confirms its robustness against both external eavesdropping attacks and internal privacy threats from honest-but-curious participants. By utilizing single-photon encoding, rotation-based encryption, and single-particle measurement, the protocol achieves enhanced scalability and practical feasibility for multi-element comparisons, advancing beyond the scope of existing single-integer comparison schemes.

References

[1]	C. Portmann, R. Renner, Security in quantum cryptography, Rev. Mod. Phys., 94 (2022), 025008. https://doi.org/10.1103/RevModPhys.94.025008 doi: 10.1103/RevModPhys.94.025008
[2]	C. H. Bennett, G. Brassard, Quantum cryptography: public key distribution and coin tossing, Theor. Comput. Sci., 560 (2014), 7–11. https://doi.org/10.1016/j.tcs.2014.05.025 doi: 10.1016/j.tcs.2014.05.025
[3]	Y. Qin, J. Cheng, J. Ma, D. Zhao, Z. Yan, X. Jia, et al., Efficient and secure quantum secret sharing for eight users, Phys. Rev. Res., 6 (2024), 033036. https://doi.org/10.1103/PhysRevResearch.6.033036 doi: 10.1103/PhysRevResearch.6.033036
[4]	A. Di Santo, W. Tiberti, D. Cassioli, Security and fairness in multi-party quantum secret sharing protocol, IEEE Trans. Quantum Eng., 6 (2025), 4100518. https://doi.org/10.1109/TQE.2025.3535823 doi: 10.1109/TQE.2025.3535823
[5]	Y.-B. Sheng, L. Zhou, G.-L. Long, One-step quantum secure direct communication, Sci. Bull., 67 (2022), 367–374. https://doi.org/10.1016/j.scib.2021.11.002 doi: 10.1016/j.scib.2021.11.002
[6]	H. Zhang, Z. Sun, R. Qi, L. Yin, G.-L. Long, J. Yu, Realization of quantum secure direct communication over 100 km fiber with time-bin and phase quantum states, Light Sci. Appl., 11 (2022), 83. https://doi.org/10.1038/s41377-022-00769-w doi: 10.1038/s41377-022-00769-w
[7]	Y.-X. Shu, C.-M. Bai, S. Zhang, Multiparty quantum key agreement based on GHZ states, EPJ Quantum Technol., 12 (2025): 48. https://doi.org/10.1140/epjqt/s40507-025-00353-2 doi: 10.1140/epjqt/s40507-025-00353-2
[8]	R.-H. Shi, Y.-F. Li, Quantum private set intersection cardinality protocol with application to privacy-preserving condition query, IEEE Trans. Circuits Syst. I Regul. Pap., 69 (2022), 2399–2411. https://doi.org/10.1109/TCSI.2022.3152591 doi: 10.1109/TCSI.2022.3152591
[9]	R.-H. Shi, Y.-F. Li, Quantum protocol for secure multiparty logical AND with application to multiparty private set intersection cardinality, IEEE Trans. Circuits Syst. I Regul. Pap., 69 (2022), 5206–5218. https://doi.org/10.1109/TCSI.2022.3200974 doi: 10.1109/TCSI.2022.3200974
[10]	M. Hou, Y. Wu, S. Zhang, Quantum private set intersection scheme based on Bell states, Axioms, 14 (2025), 120. https://doi.org/10.3390/axioms14020120 doi: 10.3390/axioms14020120
[11]	D. Feng, K. Yang, Concretely efficient secure multi-party computation protocols: survey and more, Security and Safety, 1 (2022), 2021001. https://doi.org/10.1051/sands/2021001 doi: 10.1051/sands/2021001
[12]	P. Tamilselvi, V. Lathika, S. Jayachitra, S Arunkumar, M. Balasubramani, V Kalaichelvi, Secure multi-party computation for collaborative data analysis in network security, In: 2024 International conference on intelligent and innovative technologies in computing, electrical and electronics (IITCEE), Bangalore, India, 2024, 1–5. https://doi.org/10.1109/IITCEE59897.2024.10467913
[13]	X. Qian, L. Wei, J. Zhang, L. Zhang, Malicious-secure threshold multi-party private set intersection for anonymous electronic voting, Cryptography, 9 (2025), 23. https://doi.org/10.3390/cryptography9020023 doi: 10.3390/cryptography9020023
[14]	I. Zhou, F. Tofigh, M. Piccardi, M. Abolhasan, D. Franklin, J. Lipman, Secure multi-party computation for machine learning: a survey, IEEE Access, 12 (2024), 53881–53899. https://doi.org/10.1109/ACCESS.2024.3388992 doi: 10.1109/ACCESS.2024.3388992
[15]	Y.-Y. Li, F.-C. Qian, G.-R. Zhang, X.-C. Li, L.-W. Zhou, Z.-M. Yu, et al., FunlncModel: integrating multi-omic features from upstream and downstream regulatory networks into a machine learning framework to identify functional lncRNAs, Brief. Binioform., 26 (2025), bbae623. https://doi.org/10.1093/bib/bbae623 doi: 10.1093/bib/bbae623
[16]	A. C. Yao, Protocols for secure computations, In: 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982), Chicago, IL, USA, 1982,160–164. https://doi.org/10.1109/SFCS.1982.38
[17]	F. Boudot, B. Schoenmakers, J. Traoré, A fair and efficient solution to the socialist millionaires' problem, Discrete Appl. Math., 111 (2001), 23–36. https://doi.org/10.1016/S0166-218X(00)00342-5 doi: 10.1016/S0166-218X(00)00342-5
[18]	H.-K. Lo, Insecurity of quantum secure computations, Phys. Rev. A, 56 (1997), 1154–1162. https://doi.org/10.1103/PhysRevA.56.1154 doi: 10.1103/PhysRevA.56.1154
[19]	P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Rev., 41 (1999), 303–332. https://doi.org/10.1137/S0036144598347011 doi: 10.1137/S0036144598347011
[20]	L. K. Grover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett., 79 (1997), 325–328. https://doi.org/10.1103/PhysRevLett.79.325 doi: 10.1103/PhysRevLett.79.325
[21]	Y.-G. Yang, Q.-Y. Wen, An efficient two-party quantum private comparison protocol with decoy photons and two-photon entanglement, J. Phys. A: Math. Theor., 42 (2009), 055305. https://doi.org/10.1088/1751-8113/42/5/055305 doi: 10.1088/1751-8113/42/5/055305
[22]	B. Liu, D. Xiao, W. Huang, H.-Y. Jia, T.-T. Song, Quantum private comparison employing single-photon interference, Quantum Inf. Process., 16 (2017), 180. https://doi.org/10.1007/s11128-017-1630-y doi: 10.1007/s11128-017-1630-y
[23]	H.-M. Pan, Two-party quantum private comparison using single photons, Int. J. Theor. Phys., 57 (2018), 3389–3395. https://doi.org/10.1007/s10773-018-3852-x doi: 10.1007/s10773-018-3852-x
[24]	J.-W. Zhang, G. Xu, X.-B. Chen, Y. Chang, Z.-C. Dong, Improved multiparty quantum private comparison based on quantum homomorphic encryption, Physica A, 610 (2023), 128397. https://doi.org/10.1016/j.physa.2022.128397 doi: 10.1016/j.physa.2022.128397
[25]	M. Hou, Y. Wu, New quantum private comparison using Bell states, Entropy, 26 (2024), 682. https://doi.org/10.3390/e26080682 doi: 10.3390/e26080682
[26]	Y.-F. Lang, Quantum private comparison using single bell state, Int. J. Theor. Phys., 60 (2021), 4030–4036. https://doi.org/10.1007/s10773-021-04937-3 doi: 10.1007/s10773-021-04937-3
[27]	X. Huang, S. B. Zhang, Y. Chang, M. Hou, W. Cheng, Efficient quantum private comparison based on entanglement swapping of bell states, Int. J. Theor. Phys., 60 (2021), 3783–3796. https://doi.org/10.1007/s10773-021-04915-9 doi: 10.1007/s10773-021-04915-9
[28]	T.-Y. Ye, Z.-X. Ji, Two-party quantum private comparison with five-qubit entangled states, Int. J. Theor. Phys., 56 (2017), 1517–1529. https://doi.org/10.1007/s10773-017-3291-0 doi: 10.1007/s10773-017-3291-0
[29]	Q. Sun, Quantum private comparison with six-particle maximally entangled states, Mod. Phys. Lett. A, 37 (2022), 2250149. https://doi.org/10.1142/S0217732322501498 doi: 10.1142/S0217732322501498
[30]	C. Li, X. Chen, H. Li, Y. Yang, J. Li, Efficient quantum private comparison protocol based on the entanglement swapping between four-qubit cluster state and extended Bell state, Quantum Inf. Process., 18 (2019), 158. https://doi.org/10.1007/s11128-019-2266-x doi: 10.1007/s11128-019-2266-x
[31]	M. Hou, Y. Wu, Quantum private comparison protocol with cluster states, Axioms, 14 (2025), 70. https://doi.org/10.3390/axioms14010070 doi: 10.3390/axioms14010070
[32]	S. Lin, Y. Sun, X.-F. Liu, Z.-Q. Yao, Quantum private comparison protocol with d-dimensional Bell states, Quantum Inf. Process., 12 (2013), 559–568. https://doi.org/10.1007/s11128-012-0395-6 doi: 10.1007/s11128-012-0395-6
[33]	F. Z. Guo, F. Gao, S. J. Qin, J. Zhang, Q. Y. Wen, Quantum private comparison protocol based on entanglement swapping of d-level Bell states, Quantum Inf. Process., 12 (2013), 2793–2802. https://doi.org/10.1007/s11128-013-0536-6 doi: 10.1007/s11128-013-0536-6
[34]	W. Q. Wu, Y. X. Zhao, Quantum private comparison of size using d-level Bell states with a semi-honest third party, Quantum Inf. Process., 20 (2021), 155. https://doi.org/10.1007/s11128-021-03059-3 doi: 10.1007/s11128-021-03059-3
[35]	Y.-F. Lang, Quantum private magnitude comparison, Int. J. Theor. Phys., 61 (2022), 100. https://doi.org/10.1007/s10773-022-05043-8 doi: 10.1007/s10773-022-05043-8
[36]	M. Hou, Y. Wu, Two-party quantum private comparison protocol for direct secret comparison, Mathematics, 13 (2025), 326. https://doi.org/10.3390/math13020326 doi: 10.3390/math13020326
[37]	Y.-F. Hu, L.-H. Gong, Q.-W. Zeng, Efficient and robust multi-party semi-quantum private comparison protocols with decoherence-free states over the collective noises channel, Quantum Inf. Process., 24 (2025), 216. https://doi.org/10.1007/s11128-025-04832-4 doi: 10.1007/s11128-025-04832-4
[38]	C.-Q. Ye, J. Li, X.-B. Chen, Y. Hou, A feasible semi-quantum private comparison based on entanglement swapping of Bell states, Physica A, 625 (2023), 129023. https://doi.org/10.1016/j.physa.2023.129023 doi: 10.1016/j.physa.2023.129023
[39]	J.-H. Huang, M.-L. Li, Y.-Y. Liu, L.-G. Qin, L.-H. Gong, Efficient semi-quantum private comparison protocol of size relation based on high dimensional Bell states, Chinese Phys. B, in press. https://doi.org/10.1088/1674-1056/ae2abd
[40]	X. Huang, W. Zhang, X. Wang, S. Zhang, M. K. Khan, QF2PM: quantum-secure fine-grained privacy-preserving profile matching for mobile social networks, IEEE Trans. Netw. Sci. Eng., 13 (2026), 2678–2693. https://doi.org/10.1109/TNSE.2025.3619531 doi: 10.1109/TNSE.2025.3619531
[41]	E. Campbell, A series of fast-paced advances in quantum error correction, Nat. Rev. Phys., 6 (2024), 160–161. https://doi.org/10.1038/s42254-024-00706-3 doi: 10.1038/s42254-024-00706-3
[42]	V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, et al., Real-time quantum error correction beyond break-even, Nature, 616 (2023), 50–55. https://doi.org/10.1038/s41586-023-05782-6 doi: 10.1038/s41586-023-05782-6
[43]	Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature, 638 (2025), 920–926. https://doi.org/10.1038/s41586-024-08449-y doi: 10.1038/s41586-024-08449-y
[44]	X. Huang, W. Zhang, S. Zhang, M. K. Khan, Quantum-secure privacy-preserving profile matching for proximity-based mobile social networks, IEEE Trans. Consum. Electr., in press. https://doi.org/10.1109/TCE.2026.3666093

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)