Privacy-preserving computation meets quantum computing: A scoping review^☆

Abstract

Privacy-Preserving Computation (PPC) comprises the techniques, schemes and protocols which ensure privacy and confidentiality in the context of secure computation and data analysis. Most of the current PPC techniques rely on the complexity of cryptographic operations, which are expected to be efficiently solved by quantum computers soon. This review explores how PPC can be built on top of quantum computing itself to alleviate these future threats. We analyze quantum proposals for Secure Multi-party Computation, Oblivious Transfer and Homomorphic Encryption from the last decade focusing on their maturity and the challenges they currently face. Our findings show a strong focus on purely theoretical works, but a rise on the experimental consideration of these techniques in the last 5 years. The applicability of these techniques to actual use cases is an underexplored aspect which could lead to the practical assessment of these techniques.

Keywords

Quantum computing / Privacy-preserving computation / Oblivious transfer / Secure multi-party computation / Homomorphic encryption / Scoping review

References

[1] P. Mohassel, Y. Zhang,SecureML: a system for scalable privacy-preserving machine learning, in: 2017 IEEE Symposium on Security and Privacy (SP), 2017, pp. 19-38.

[2] C. Zhao, S. Zhao, M. Zhao, Z. Chen, C.-Z. Gao, H. Li, Y.-a. Tan, Secure multi-party computation: theory, practice and applications, Inf. Sci. 476 (2019) 357-372.

[3] V.K. Yadav, N. Andola, S. Verma, S. Venkatesan, A survey of oblivious transfer protocol, ACM Comput. Surv. 54 (10s) (2022) 211:1-211:37.

[4] A.C. Tricco, E. Lillie, W. Zarin, K.K. O’Brien, H. Colquhoun, D. Levac, D. Moher, M. D. Peters, T. Horsley, L. Weeks, S. Hempel, E.A. Akl, C. Chang, J. McGowan, L. Stewart, L. Hartling, A. Aldcroft, M.G. Wilson, C. Garritty, S. Lewin, C.M. God-frey, M.T. Macdonald, E.V. Langlois, K. Soares-Weiser, J. Moriarty, T. Clifford, O. Tunçalp, S.E. Straus, PRISMA extension for scoping reviews (PRISMA-ScR): check-list and explanation, Ann. Intern. Med. 169 (7) (2018) 467-473.

[5] J. Preskill, Quantum Computing 40 Years Later, 2nd edition, CRC Press, 2023.

[6] S. Wiesner, Conjugate coding, ACM SIGACT News 15 (1) (1983) 78-88.

[7] C.H. Bennett, G. Brassard, Quantum cryptography: Public key distribution and coin tossing, 1984, pp. 175-179.

[8] V. Scarani, H. Bechmann-Pasquinucci, N.J. Cerf, M. Dušek, N. Lütkenhaus, M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys. 81 (3) (2009) 1301-1350.

[9] A. Brazaola-Vicario, A. Ruiz, O. Lage, E. Jacob, J. Astorga, Quantum key distribu-tion: a survey on current vulnerability trends and potential implementation risks, Opt. Contin. 3 (8) (2024) 1438-1460.

[10] P. Shor, Algorithms for quantum computation: discrete logarithms and factoring,in: Proceedings 35th Annual Symposium on Foundations of Computer Science, 1994, pp. 124-134.

[11] D.J. Bernstein, T. Lange, Post-quantum cryptography, Nature 549 (7671) (2017) 188-194.

[12] L.K. Grover, A fast quantum mechanical algorithm for database search, in: Proceed-ings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, Association for Computing Machinery, New York, NY, USA, 1996, pp. 212-219.

[13] A.C. Yao, Protocols for secure computations, in: 23rd Annual Symposium on Foun-dations of Computer Science (sfcs 1982), 1982, pp. 160-164.

[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.

[15] F. Kerschbaum, Privacy-preserving computation, in: B. Preneel, D. Ikonomou (Eds.), Privacy Technologies and Policy, in: Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 2014, pp. 41-54.

[16] A. Shamir, How to share a secret, Commun. ACM 22 (11) (1979) 612-613.

[17] A.C.-C. Yao, How to generate and exchange secrets, in: 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), 1986, pp. 162-167.

[18] M. Bellare, V.T. Hoang, S. Keelveedhi, P. Rogaway, Efficient garbling from a fixed-key blockcipher, in: 2013 IEEE Symposium on Security and Privacy, 2013, pp. 478-492.

[19] S. Gueron, Y. Lindell, A. Nof, B. Pinkas, Fast garbling of circuits under standard assumptions, in: Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, CCS ’15, Association for Computing Machinery, New York, NY, USA, 2015, pp. 567-578.

[20] C. Guo, J. Katz, X. Wang, Y. Yu, Efficient and secure multiparty computation from fixed-key block ciphers, in: 2020 IEEE Symposium on Security and Privacy (SP), 2020, pp. 825-841.

[21] M.O. Rabin, How to Exchange Secrets with Oblivious Transfer, Tech. Rep. 2005/187, Harvard University, Aiken Computation Lab, May 1981.

[22] M. Jakobi, C. Simon, N. Gisin, J.-D. Bancal, C. Branciard, N. Walenta, H. Zbinden, Practical private database queries based on a quantum-key-distribution protocol, Phys. Rev. A 83 (2) (2011) 022301.

[23] Y. Gertner, S. Kannan, T. Malkin, O. Reingold, M. Viswanathan,The relationship between public key encryption and oblivious transfer, in:Proceedings 41st Annual Symposium on Foundations of Computer Science, 2000, pp. 325-335.

[24] R.L. Rivest, L. Adleman, M.L. Dertouzos, On data banks and privacy homomor-phisms, in: Foundations of Secure Computation 4 ( 11), 1978, pp. 169-180.

[25] R.L. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM 21 (2) (1978) 120-126.

[26] C. Gentry, Fully homomorphic encryption using ideal lattices, in: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, STOC ’09, Asso-ciation for Computing Machinery, New York, NY, USA, 2009, pp. 169-178.

[27] A. Broadbent, C. Schaffner, Quantum cryptography beyond quantum key distribu-tion, Des. Codes Cryptogr. 78 (1) (2016) 351-382.

[28] F. Gao, S. Qin, W. Huang, Q. Wen,Quantum private query: a new kind of practical quantum cryptographic protocol, Sci. China, Phys. Mech. Astron. 62 (2019) 70301.

[29] M.B. Santos, P. Mateus, A.N. Pinto, Quantum oblivious transfer: a short review, Entropy 24 (7) (2022) 945.

[30] J.F. Fitzsimons, Private quantum computation: an introduction to blind quantum computing and related protocols, npj Quantum Inf. 3 (1) (2017) 1-11.

[31] A. Martín-Martín, M. Thelwall, E. Orduna-Malea, E. Delgado López-Cózar, Google scholar, Microsoft academic scopus, dimensions, web of science, and OpenCita-tions’ COCI: a multidisciplinary comparison of coverage via citations, Scientomet-rics 126 (1) (2021) 871-906.

[32] Scopus, https://www.scopus.com, 2024. (Accessed 12 March 2024).

[33] Zotero, https://www.zotero.org, 2024. (Accessed 10 May 2024).

[34] P. Chan, I. Lucio-Martinez, X. Mo, C. Simon, W. Tittel, Performing private database queries in a real-world environment using a quantum protocol, Sci. Rep. 4 (2014) 5233.

[35] Y.-G. Yang, P. Xu, J. Tian, H. Zhang, Quantum oblivious transfer with an untrusted third party, Optik 125 (18) (2014) 5409-5413.

[36] Y.-G. Yang, S.-J. Sun, Y. Wang, Quantum oblivious transfer based on a quantum symmetrically private information retrieval protocol, Int. J. Theor. Phys. 54 (3)(2015) 910-916.

[37] W. Liu, P. Gao, Z. Liu, H. Chen, M. Zhang, A quantum-based database query scheme for privacy preservation in cloud environment, Secur. Commun. Netw. (2019).

[38] H. Xiao, W.-H. Huang, M. Zhou, An efficient quantum private query protocol based on oracle and Grover iteration, Int. J. Theor. Phys. 58 (9) (2019) 3025-3035.

[39] J. Zhao, W. Zhang, Y. Ma, X. Zhang, H. Ma, Development of quantum private queries protocol on collective-dephasing noise channel, App. Sci. (Switzerland) 10 (6) (2020) 1935.

[40] S. Song, M. Hayashi, Capacity of quantum private information retrieval with mul-tiple servers, IEEE Trans. Inf. Theory 67 (1) (2021) 452-463.

[41] H.-K. Lo, Insecurity of quantum secure computations, Phys. Rev. A, At. Mol. Opt. Phys. 56 (2) (1997) 1154-1162.

[42] R. Colbeck, Impossibility of secure two-party classical computation, Phys. Rev. A, At. Mol. Opt. Phys. 76 (6) (2007).

[43] S. Kundu, J. Sikora, E.-Z. Tan, A device-independent protocol for XOR oblivious transfer, Quantum 6 (2022) 725.

[44] A. Broadbent, P. Yuen, Device-independent oblivious transfer from the bounded-quantum-storage-model and computational assumptions, New J. Phys. 25 (5)(2023).

[45] D. Pitalúa-García, Spacetime-constrained oblivious transfer, Phys. Rev. A 93 (6)(2016).

[46] D. Pitalúa-García, I. Kerenidis, Practical and unconditionally secure spacetime-constrained oblivious transfer, Phys. Rev. A 98 (3) (2018).

[47] D. Pitalúa-García, One-out-of-m spacetime-constrained oblivious transfer, Phys. Rev. A 100 (1) (2019).

[48] L. Salvail, C. Schaffner, M. Sotáková, Quantifying the leakage of quantum protocols for classical two-party cryptography, Int. J. Quantum Inf. 12 (6) (2015).

[49] S. Winkler, J. Wullschleger, On the efficiency of classical and quantum secure func-tion evaluation, IEEE Trans. Inf. Theory 60 (6) (2014) 3123-3143.

[50] G. He, Secure quantum weak oblivious transfer against individual measurements, Quantum Inf. Process. 14 (6) (2015) 2153-2170.

[51] Y.-G. Yang, R. Yang, H. Lei, W.-M. Shi, Y.-H. Zhou, Quantum oblivious trans-fer with relaxed constraints on the receiver, Quantum Inf. Process. 14 (8) (2015) 3031-3040.

[52] R. Amiri, R. Stárek, D. Reichmuth, I. Puthoor, M. Mičuda, L. Mišta, M. Dušek, P. Wallden, E. Andersson, Imperfect 1-Out-of-2 quantum oblivious transfer: bounds, a protocol, and its experimental implementation, PRX Quantum 2 (1) (2021).

[53] K. Zhang, C. Ma, Z. Sun, X. Zhang, B. Zhou, Y. Wang, Privacy-preserving decision protocols based on quantum oblivious key distribution, Comput. Mater. Continua 64 (3) (2020) 1915-1928.

[54] G. He, An optical implementation of quantum bit commitment using infinite-dimensional systems, Appl. Sci. (Switzerland) 13 (13) (2023).

[55] R.-H. Shi, Y.-F. Li, Quantum protocol for secure multiparty logical AND with appli-cation to multiparty private set intersection cardinality, IEEE Trans. Circuits Syst. I, Regul. Pap. 69 (12) (2022) 5206-5218.

[56] X.-Q. Fang, R.-H. Shi, Cloud-assisted quantum primitive protocols and applications, Phys. Scr. 98 (9) (2023).

[57] R.-H. Shi, Useful equations about bell states and their applications to quantum secret sharing, IEEE Commun. Lett. 24 (2) (2020) 386-390.

[58] S.-J. Fu, K.-J. Zhang, L. Zhang, K.-C. Hou, A new non-entangled quantum secret sharing protocol among different nodes in further quantum networks, Front. Phys. 10 (2022).

[59] T. Wu, S. Zhu, F. Li, L. Liu, Two Quantum Secret Sharing Schemes with Adversary Structure, Int. J. Theor. Phys. 61 (7) (2022).

[60] R.-H. Shi, Quantum sealed-bid auction without a trusted third party, IEEE Trans. Circuits Syst. I, Regul. Pap. 68 (10) (2021) 4221-4231.

[61] R.-H. Shi, A generic quantum protocol for one-sided secure two-party classical com-putations, Quantum Inf. Process. 19 (1) (2020).

[62] C. Lu, F. Miao, J. Hou, Z. Su, Y. Xiong, Secure multi-party computation with a quantum manner, J. Phys. A, Math. Theor. 54 (8) (2021).

[63] Z. Deng, Y. Zhang, X. Zhang, L. Li, Privacy-preserving quantum multi-party com-putation based on circular structure, J. Inf. Secur. Appl. 47 (2019) 120-124.

[64] H. Abulkasim, A. Mashatan, S. Ghose, Security improvements for privacy-preserving quantum multiparty computation based on circular structure, Quantum Inf. Process. 21 (1) (2022).

[65] G.-J. Qu, M.-M. Wang, Secure multi-party quantum computation based on blind quantum computation, Int. J. Theor. Phys. 60 (8) (2021) 3003-3012.

[66] M. Clementi, A. Pappa, A. Eckstein, I. Walmsley, E. Kashefi, S. Barz, Classical mul-tiparty computation using quantum resources, Phys. Rev. A 96 (6) (2017).

[67] H. Cao, W. Ma, G. Liu, L. Lü, Z.-Y. Xue, Quantum Secure Multiparty Computation with Symmetric Boolean Functions, Chin. Phys. Lett. 37 (5) (2020).

[68] S. Debnath, K. Dey, N. Kundu, T. Choudhury, Feasible private set intersection in quantum domain, Quantum Inf. Process. 20 (1) (2021).

[69] Y. Wang, P. Hu, Q. Xu, Quantum protocols for private set intersection cardinality and union cardinality based on entanglement swapping, Int. J. Theor. Phys. 60 (9)(2021) 3514-3528.

[70] R.-H. Shi, Quantum multiparty privacy set intersection cardinality, IEEE Trans. Cir-cuits Syst. II, Express Briefs 68 (4) (2021) 1203-1207.

[71] W.-J. Liu, W.-B. Li, H.-B. Wang, An improved quantum private set intersection pro-tocol based on Hadamard gates, Int. J. Theor. Phys. 61 (3) (2022).

[72] R.-H. Shi, X.-Q. Fang, Edge-assisted quantum protocol for secure multiparty logical AND its applications, iScience 26 (7) (2023).

[73] W. Liu, Q. Yang, Z. Li, Quantum multi-party private set union protocol based on least common multiple and Shor’s algorithm, Int. J. Quantum Inf. 21 (7) (2023).

[74] R.-H. Shi, Y. Mu, H. Zhong, J. Cui, S. Zhang, Two Quantum Protocols for Oblivious Set-member Decision Problem, Sci. Rep. 5 (2015).

[75] R.-H. Shi, Efficient quantum protocol for private set intersection cardinality, IEEE Access 6 (2018) 73102-73109.

[76] R.-H. Shi, M. Zhang, A feasible quantum protocol for private set intersection cardi-nality, IEEE Access 7 (2019) 72105-72112.

[77] B. Liu, M. Zhang, R. Shi, Quantum secure multi-party private set intersection car-dinality, Int. J. Theor. Phys. 59 (7) (2020) 1992-2007.

[78] R.-H. Shi, Y.-F. Li, Quantum private set intersection cardinality protocol with ap-plication to privacy-preserving condition query, IEEE Trans. Circuits Syst. I, Regul. Pap. 69 (6) (2022) 2399-2411.

[79] G. Xu, F. Yun, X.-B. Chen, S. Xu, J. Wang, T. Shang, Y. Chang, M. Dong, Secure multi-party quantum summation based on quantum homomorphic encryption, In-tell. Autom. Soft Comput. 34 (1) (2022) 531-541.

[80] C.-W. Yang, C.-W. Tsai, C.-A. Chen, J. Lin, Robust semi-quantum summation over a collective-dephasing noise channel, Mathematics 11 (6) (2023).

[81] C. Zhang, Y. Long, Q. Li, Quantum summation using d-level entanglement swap-ping, Quantum Inf. Process. 20 (4) (2021).

[82] X.-Q. Cai, T.-Y. Wang, C.-Y. Wei, F. Gao, Cryptanalysis of secure multiparty quan-tum summation, Quantum Inf. Process. 21 (8) (2022).

[83] Y. Wang, P. Hu, Q. Xu, Quantum secure multi-party summation based on entangle-ment swapping, Quantum Inf. Process. 20 (10) (2021).

[84] Z. Ji, H. Zhang, H. Wang, F. Wu, J. Jia, W. Wu, Quantum protocols for secure multi-party summation, Quantum Inf. Process. 18 (6) (2019).

[85] X.-B. Chen, Y. Su, G. Xu, Y. Sun, Y.-X. Yang, Quantum state secure transmission in network communications, Inf. Sci. 276 (2014) 363-376.

[86] X.-Y. Chang, D.-L. Deng, X.-X. Yuan, P.-Y. Hou, Y.-Y. Huang, L.-M. Duan, Experi-mental realization of an entanglement access network and secure multi-party com-putation, Sci. Rep. 6 (2016).

[87] W. Liu, Y.-B. Wang, W.-Q. Fan, An novel protocol for the quantum secure multi-party summation based on two-particle bell states, Int. J. Theor. Phys. 56 (9) (2017) 2783-2791.

[88] T.-Y. Ye, J.-L. Hu, Quantum secure multiparty summation based on the phase shift-ing operation of d-level quantum system and its application, Int. J. Theor. Phys. 60 (3) (2021) 819-827.

[89] M. Hayashi, T. Koshiba, Quantum verifiable protocol for secure modulo zero-sum randomness, Quantum Inf. Process. 21 (8) (2022).

[90] R.-H. Shi, Y. Mu, H. Zhong, J. Cui, S. Zhang, Secure Multiparty Quantum Compu-tation for Summation and Multiplication, Sci. Rep. 6 (2016).

[91] Z. Dou, G. Xu, X.-B. Chen, X.-X. Niu, Y.-X. Yang, Rational protocol of quantum secure multi-party computation, Quantum Inf. Process. 17 (8) (2018).

[92] Z. Dou, X.-B. Chen, G. Xu, W. Liu, Y.-X. Yang, Y. Yang, An attempt at universal quantum secure multi-party computation with graph state, Phys. Scr. 95 (5) (2020).

[93] X. Song, R. Gou, A. Wen, Secure multiparty quantum computation based on La-grange unitary operator, Sci. Rep. 10 (1) (2020).

[94] K. Sutradhar, H. Om, An efficient simulation for quantum secure multiparty com-putation, Sci. Rep. 11 (1) (2021).

[95] Z.-W. Peng, R.-H. Shi, H. Zhong, J. Cui, S. Zhang, A novel quantum scheme for secure two-party distance computation, Quantum Inf. Process. 16 (12) (2017).

[96] B. Chen, W. Yang, L. Huang, Cryptanalysis and improvement of the novel quantum scheme for secure two-party distance computation, Quantum Inf. Process. 18 (1)(2019).

[97] Z.-W. Peng, R.-H. Shi, P.-H. Wang, S. Zhang, Two quantum protocols for secure Hamming distance computation, Quantum Inf. Process. 18 (1) (2019).

[98] Y. Cao, Quantum secure two-party Euclidean distance computation based on mu-tually unbiased bases, Quantum Inf. Process. 21 (7) (2022).

[99] W. Liu, Y. Xu, J. Yang, W. Yu, L. Chi, Privacy-preserving quantum two-party geo-metric intersection, Comput. Mater. Continua 60 (3) (2019) 1237-1250.

[100] Z.-X. Li, Q. Yang, B. Feng, W.-J. Liu, Quantum Privacy-preserving Two-party Circle Intersection Protocol Based on Phase-encoded Query, Int. J. Theor. Phys. 62 (7)(2023).

[101] R.-H. Shi, Y. Mu, H. Zhong, J. Cui, S. Zhang, Privacy-preserving point-inclusion protocol for an arbitrary area based on phase-encoded quantum private query, Quantum Inf. Process. 16 (1) (2017).

[102] C. Wang, R.-G. Zhou, Secure multi-party convex hull protocol based on quantum homomorphic encryption, Quantum Inf. Process. 22 (1) (2023).

[103] Q.-L. Wang, H.-X. Sun, W. Huang, Multi-party quantum private comparison proto-col with n-level entangled states, Quantum Inf. Process. 13 (11) (2014) 2375-2389.

[104] Y.-H. Zhou, W.-M. Shi, Y.-G. Yang, Comment on “efficient and feasible quantum private comparison of equality against the collective amplitude damping noise”, Quantum Inf. Process. 13 (2) (2014) 573-585.

[105] K. Thapliyal, R. Sharma, A. Pathak, Orthogonal-state-based and semi-quantum pro-tocols for quantum private comparison in noisy environment, Int. J. Quantum Inf. 16 (5) (2018).

[106] L. Li, R.-H. Shi, A novel and efficient quantum private comparison scheme, J. Ko-rean Phys. Soc. 75 (1) (2019) 15-21.

[107] T.-Y. Kou, B.-C. Che, Z. Dou, X.-B. Chen, Y.-P. Lai, J. Li, Efficient quantum private comparison protocol utilizing single photons and rotational encryption, Chin. Phys. B 31 (6) (2022).

[108] P. Fan, A. Rahman, Z. Ji, X. Ji, Z. Hao, H. Zhang, Two-party quantum private com-parison based on eight-qubit entangled state, Mod. Phys. Lett. A 37 (5) (2022).

[109] W.-W. Zhang, D. Li, Y.-B. Li, Quantum private comparison protocol with W states, Int. J. Theor. Phys. 53 (5) (2014) 1723-1729.

[110] J. Joseph, S. Ali, Multiparty quantum private comparison based on quantum walks, Int. J. Quantum Inf. 22 (1) (2023).

[111] W. Liu, Y.-B. Wang, A.-N. Sui, M.-Y. Ma, Quantum protocol for millionaire problem, Int. J. Theor. Phys. 58 (7) (2019) 2106-2114.

[112] A. Saxena, K. Thapliyal, A. Pathak, Continuous variable controlled quantum di-alogue and secure multiparty quantum computation, Int. J. Quantum Inf. 18 (4)(2020).

[113] W.-M. Shi, S.-S. Liu, Y.-H. Zhou, Y.-G. Yang, A secure quantum multi-party ranking protocol based on continuous variables, Optik 241 (2021).

[114] Q.-B. Luo, X.-Y. Li, G.-W. Yang, C. Lin, A mediated semi-quantum protocol for millionaire problem based on high-dimensional Bell states, Quantum Inf. Process. 21 (7) (2022).

[115] Y. Zhang, L. Zhang, K. Zhang, W. Wang, K. Hou, A new quantum-inspired solution to blind millionaires’ problem, Quantum Inf. Process. 22 (1) (2023).

[116] R.-H. Shi, Verifiable quantum encryption and its practical applications, Int. J. Theor. Phys. 56 (4) (2017) 1208-1217.

[117] R. Christensen, P. Popovski, Private Product Computation Using Quantum Entan-glement, IEEE Trans. Quantum Eng. 4 (2023).

[118] W.-J. Liu, Z.-X. Li, Secure and efficient two-party quantum scalar product protocol with application to privacy-preserving matrix multiplication, IEEE Trans. Circuits Syst. I, Regul. Pap. 70 (11) (2023) 4456-4469.

[119] Q. Wang, C. Yu, F. Gao, H. Qi, Q. Wen, Self-tallying quantum anonymous voting, Phys. Rev. A 94 (2) (2016).

[120] L. Chen, L. Chen, Q. Li, Practical multi-party quantum homomorphic encryption, Theor. Comput. Sci. 971 (2023).

[121] D. Pitalúa-García,Unconditionally secure relativistic multi-party biased coin flip-ping and die rolling, Proc. Royal Soc. A, Math. Phys. Eng. Sci. 477 (2252) (2021).

[122] M. Liang, Quantum fully homomorphic encryption scheme based on universal quantum circuit, Quantum Inf. Process. 14 (8) (2015) 2749-2759.

[123] S.-H. Tan, J. Kettlewell, Y. Ouyang, L. Chen, J. Fitzsimons, A quantum approach to homomorphic encryption, Sci. Rep. 6 (2016).

[124] X. Sun, T. Wang, Z. Sun, P. Wang, J. Yu, W. Xie, An efficient quantum somewhat homomorphic symmetric searchable encryption, Int. J. Theor. Phys. 56 (4) (2017) 1335-1345.

[125] C. Gong, J. Du, Z. Dong, Z. Guo, A. Gani, L. Zhao, H. Qi, Grover algorithm-based quantum homomorphic encryption ciphertext retrieval scheme in quantum cloud computing, Quantum Inf. Process. 19 (3) (2020).

[126] Y. Ouyang, S.-H. Tan, J. Fitzsimons, P. Rohde, Homomorphic encryption of linear optics quantum computation on almost arbitrary states of light with asymptotically perfect security, Phys. Rev. Res. 2 (1) (2020).

[127] J.-W. Zhang, X.-B. Chen, G. Xu, Y.-X. Yang, Universal quantum circuit evaluation on encrypted data using probabilistic quantum homomorphic encryption scheme, Chin. Phys. B 30 (7) (2021).

[128] Y. Zhang, T. Shang, J. Liu, A multi-valued quantum fully homomorphic encryption scheme, Quantum Inf. Process. 20 (3) (2021).

[129] J. Liu, Q. Li, J. Quan, C. Wang, J. Shi, H. Situ, Efficient quantum homomorphic encryption scheme with flexible evaluators and its simulation, Des. Codes Cryptogr. 90 (3) (2022) 577-591.

[130] U. Mahadev, Classical homomorphic encryption for quantum circuits, SIAM J. Com-put. 52 (6) (2023) FOCS18189-FOCS18215.

[131] W. Chang, Z.-Z. Li, F.-C. You, X.-B. Pan, Dynamic quantum fully homomorphic en-cryption scheme based on universal quantum circuit, J. Inf. Secur. Appl. 75 (2023).

[132] G. Ma, H. Li, Quantum Fully Homomorphic Encryption by Integrating Pauli One-time Pad with Quaternions, Quantum 6 (2022).

[133] J.-W. Zhang, X.-B. Chen, G. Xu, H.-J. Li, Y.-L. Wang, L.-H. Miao, Y.-X. Yang, A secure multiparty quantum homomorphic encryption scheme, Comput. Mater. Continua 73 (2) (2022) 2835-2848.

[134] C.-Y. Lai, K.-M. Chung, On statistically-secure quantum homomorphic encryption, Quantum Inf. Comput. 18 (9-10) (2018) 785-794.

[135] T. Shang, S. Wang, Y. Jiang, J. Liu, Two-round quantum homomorphic encryption scheme based on matrix decomposition, Quantum Inf. Process. 22 (12) (2023).

[136] H.-K. Lo, Insecurity of quantum secure computations, Phys. Rev. A 56 (2) (1997) 1154-1162.

[137] Y.-G. Yang, S.-J. Sun, J. Tian, P. Xu, Secure quantum private query with real-time security check, Optik 125 (19) (2014) 5538-5541.

[138] H. Yu, X. Ren, C. Zhao, S. Yang, J. McCann, Quantum-aided secure deep neural network inference on real quantum computers, Sci. Rep. 13 (1) (2023).

[139] J.-W. Zhang, G. Xu, X.-B. Chen, Y. Chang, Z.-C. Dong, Improved multiparty quan-tum private comparison based on quantum homomorphic encryption, Phys. A, Stat. Mech. Appl. 610 (2023).

[140] W. Liu, H.-W. Yin, A quantum protocol for private substitution problem, Quantum Inf. Process. 20 (5) (2021).

[141] W.-J. Liu, C.-T. Li, Y. Zheng, Y. Xu, Y.-S. Xu, Quantum privacy-preserving price E-negotiation, Int. J. Theor. Phys. 58 (10) (2019) 3259-3270.

[142] IBM Quantum Platform, https://quantum.ibm.com, 2024. (Accessed 16 April 2024).

[143] M. Santos, A. Gomes, A. Pinto, P. Mateus, Private computation of phylogenetic trees based on quantum technologies, IEEE Access 10 (2022) 38065-38088.

[144] Z. Rahmani, L. Barbosa, A. Pinto, Quantum privacy-preserving service for secure lane change in vehicular networks, IET Quantum Commun. 4 (3) (2023) 103-111.

[145] N. Li, Combination of blockchain and AI for music intellectual property protection, Comput. Intell. Neurosci. (2022).

[146] H.-X. Kong, H.-Y. Jia, X. Wu, G.-Q. Li, Robust Quantum Secure Multiparty Com-putation Protocols for Minimum Value Calculation in Collective Noises and Their Simulation, Int. J. Theor. Phys. 62 (8) (2023).

[147] Y.-H. Chou, G.-J. Zeng, S.-Y. Kuo, One-out-of-two Quantum Oblivious Transfer based on Nonorthogonal States, Sci. Rep. 8 (1) (2018).

[148] R.-H. Shi, H. Yu, Privacy-preserving range query quantum scheme with single pho-tons in edge-based Internet of things, IEEE Trans. Netw. Serv. Manag. 20 (4) (2023) 4923-4936.

[149] X. Zhang, C. Wei, S. Qin, F. Gao, Q. Wen, Practical efficient 1-out-of-n quantum oblivious transfer protocol, Quantum Inf. Process. 22 (2) (2023).

[150] X. Zhang, F. Chen, Y. Guo, F. Zhang, D. Li, Quantum Oblivious Transfer Based on Entanglement Swapping, Int. J. Theor. Phys. 62 (8) (2023).

[151] M. Santos, A. Pinto, P. Mateus, Quantum and classical oblivious transfer: a com-parative analysis, IET Quantum Commun. 2 (2) (2021) 42-53.

[152] A. Broadbent, S. Gharibian, H.-S. Zhou, Towards quantum one-time memories from stateless hardware, Quantum 5 (2021).

[153] X.-B. Chen, Y.-R. Sun, G. Xu, Y.-X. Yang, Quantum homomorphic encryption scheme with flexible number of evaluator based on (k, n)-threshold quantum state sharing, Inf. Sci. 501 (2019) 172-181.

[154] G. Chen, Y. Wang, L. Jian, Y. Zhou, S. Liu, J. Luo, K. Yang, Quantum identity authentication protocol based on flexible quantum homomorphic encryption with qubit rotation, J. Appl. Phys. 133 (6) (2023).

[155] Z.-W. Cheng, X.-B. Chen, G. Xu, Y. Chang, Y. Yang, Y.-X. Yang, A secure crossing two qubits protocol based on quantum homomorphic encryption, Quantum Sci. Technol. 7 (2) (2022).

[156] B. Costa, P. Branco, M. Goulão, M. Lemus, P. Mateus, Randomized oblivious transfer for secure multiparty computation in the quantum setting, Entropy 23 (8) (2021).

[157] Y. Dulek, C. Schaffner, F. Speelman, Quantum homomorphic encryption for polynomial-size circuits, Theory Comput. 14 (2018).

[158] C. Erven, N. Ng, N. Gigov, R. Laflamme, S. Wehner, G. Weihs, An experimental implementation of oblivious transfer in the noisy storage model, Nature Commun. 5 (2014).

[159] F. Furrer, T. Gehring, C. Schaffner, C. Pacher, R. Schnabel, S. Wehner, Continuous-variable protocol for oblivious transfer in the noisy-storage model, Nature Com-mun. 9 (1) (2018).

[160] C. Gong, Z. Dong, A. Gani, H. Qi, Quantum k-means algorithm based on trusted server in quantum cloud computing, Quantum Inf. Process. 20 (4) (2021).

[161] G. He, Can relativistic bit commitment lead to secure quantum oblivious transfer?, Eur. Phys. J. D 69 (4) (2015).

[162] G. He, Comment on “quantum oblivious transfer: a secure practical implementa-tion”, Quantum Inf. Process. 16 (4) (2017).

[163] G. He, Coherent attack on oblivious transfer based on single-qubit rotations, J. Phys. A, Math. Theor. 51 (15) (2018).

[164] G.-P. He, Coherent attacks on a practical quantum oblivious transfer protocol, Chin. Phys. B 27 (10) (2018).

[165] G. He, Practical quantum oblivious transfer with a single photon, Laser Phys. 29 (3)(2019).

[166] H.-L. Huang, Y.-W. Zhao, T. Li, F.-G. Li, Y.-T. Du, X.-Q. Fu, S. Zhang, X. Wang, W.-S. Bao, Homomorphic encryption experiments on IBM’s cloud quantum computing platform, Front. Phys. 12 (1) (2017).

[167] S.-Y. Kuo, K.-C. Tseng, Y.-H. Chou, F.-H. Tseng, Quantum oblivious transfer with reusable bell state, Comput. Mater. Continua 74 (1) (2023) 915-932.

[168] M. Lemus, M. Ramos, P. Yadav, N. Silva, N. Muga, A. Souto, N. Paunković, P. Mateus, A. Pinto, Generation and distribution of quantum oblivious keys for secure multiparty computation, Appl. Sci. (Switzerland) 10 (12) (2020) 1-11.

[169] Y.-B. Li, Q.-Y. Wen, S.-J. Qin, F.-Z. Guo, Y. Sun, Practical quantum all-or-nothing oblivious transfer protocol, Quantum Inf. Process. 13 (1) (2014) 131-139.

[170] M. Liang, Teleportation-based quantum homomorphic encryption scheme with quasi-compactness and perfect security, Quantum Inf. Process. 19 (1) (2020).

[171] W. Liu, Y. Li, Z. Wang, Y. Li, A new quantum private protocol for set intersection cardinality based on a quantum homomorphic encryption scheme for Toffoli gate, Entropy 25 (3) (2023).

[172] C. Lu, F. Miao, J. Hou, Z. Su, Y. Xiong, Quantum multiparty cryptosystems based on a homomorphic random basis encryption, Quantum Inf. Process. 19 (9) (2020).

[173] Q.-B. Luo, K.-Y. Feng, M.-H. Zheng, Quantum multi-valued Byzantine agreement based on d-dimensional entangled states, Int. J. Theor. Phys. 58 (12) (2019) 4025-4032.

[174] C. Lupo, J. Peat, E. Andersson, P. Kok, Error-tolerant oblivious transfer in the noisy-storage model, Phys. Rev. Res. 5 (3) (2023).

[175] M. Nagy, N. Nagy, Quantum oblivious transfer: a secure practical implementation, Quantum Inf. Process. 15 (12) (2016) 5037-5050.

[176] Y. Ouyang, S.-H. Tan, J. Fitzsimons, Quantum homomorphic encryption from quan-tum codes, Phys. Rev. A 98 (4) (2018).

[177] J. Ouyang, Y. Wang, Q. Li, Secure delegated quantum algorithms for solving Ma-halanobis distance, Phys. A, Stat. Mech. Appl. 625 (2023).

[178] M. Plesch, M. Pawłowski, M. Pivoluska, 1-out-of-2 oblivious transfer using a flawed bit-string quantum protocol, Phys. Rev. A 95 (4) (2017).

[179] J. Rodrigues, P. Mateus, N. Paunković, A. Souto, Oblivious transfer based on single-qubit rotations, J. Phys. A, Math. Theor. 50 (20) (2017).

[180] R.-H. Shi, B. Liu, M. Zhang, Secure two-party integer comparison protocol without any third party, Quantum Inf. Process. 20 (12) (2021).

[181] R.-H. Shi, Y.-F. Li, Privacy-preserving quantum protocol for finding the maximum value, EPJ Quantum Technol. 9 (1) (2022).

[182] R.-H. Shi, Y.-F. Li, A feasible quantum sealed-bid auction scheme without an auc-tioneer, IEEE Trans. Quantum Eng. 3 (2022).

[183] Y. Song, L. Yang, Practical quantum bit commitment protocol based on quantum oblivious transfer, Appl. Sci. (Switzerland) 8 (10) (2018).

[184] A. Souto, P. Mateus, P. Adão, N. Paunković, Reply to comment on ’Bit-string obliv-ious transfer based on quantum state computational distinguishability’, Phys. Rev. A, At. Mol. Opt. Phys. 92 (4) (2015).

[185] A. Souto, P. Mateus, P. Adão, N. Paunković, Bit-string oblivious transfer based on quantum state computational distinguishability, Phys. Rev. A, At. Mol. Opt. Phys. 91 (4) (2015).

[186] L. Stroh, N. Horová, R. Stárek, I. Puthoor, M. Mičuda, M. Dušek, E. Andersson, Noninteractive xor quantum oblivious transfer: optimal protocols and their exper-imental implementations, PRX Quantum 4 (2) (2023).

[187] S.-H. Tan, Y. Ouyang, P. Rohde, Practical somewhat-secure quantum somewhat-homomorphic encryption with coherent states, Phys. Rev. A 97 (4) (2018).

[188] W. Tham, H. Ferretti, K. Bonsma-Fisher, A. Brodutch, B. Sanders, A. Steinberg, S. Jeffery, Experimental demonstration of quantum fully homomorphic encryption with application in a two-party secure protocol, Phys. Rev. X 10 (1) (2020).

[189] D. Unruh, Everlasting multi-party computation, J. Cryptol. 31 (4) (2018) 965-1011.

[190] P. Wang, R. Zhang, G. Jiang, Z. Sun, Computationally Secure Quantum Oblivious Transfer, Adv. Quantum Technol. 5 (1) (2022).

[191] P. Wang, Y. Su, Z. Sun, All-or-nothing oblivious transfer based on the quantum one-way function, Front. Phys. 10 (2022).

[192] Y.-G. Yang, S.-J. Sun, Q.-X. Pan, P. Xu, Reductions between private informa-tion retrieval and oblivious transfer at the quantum level, Optik 126 (21) (2015) 3206-3209.

[193] Y.-G. Yang, S.-J. Sun, Q.-X. Pan, P. Xu, Quantum oblivious transfer based on unam-biguous set discrimination, Optik 126 (23) (2015) 3838-3843.

[194] Y.-G. Yang, R. Yang, W.-F. Cao, X.-B. Chen, Y.-H. Zhou, W.-M. Shi, Flexible quantum oblivious transfer, Int. J. Theor. Phys. 56 (4) (2017) 1286-1297.

[195] Y. Yao, K.-J. Zhang, T.-T. Song, L. Zhang, S.-N. Wang, The complete new solutions to the blind millionaires’ problem in d-dimensional quantum system, Phys. A, Stat. Mech. Appl. 627 (2023).

[196] J. Zeuner, I. Pitsios, S.-H. Tan, A. Sharma, J. Fitzsimons, R. Osellame, P. Walther, Experimental quantum homomorphic encryption, npj Quantum Inf. 7 (1) (2021).

[197] Q. Zhou, S. Lu, Y. Cui, L. Li, J. Sun, Quantum Search on Encrypted Data Based on Quantum Homomorphic Encryption, Sci. Rep. 10 (1) (2020).

[198] H. Zhu, C. Wang, X. Wang, Quantum fully homomorphic encryption scheme for cloud privacy data based on quantum circuit, Int. J. Theor. Phys. 60 (8) (2021) 2961-2975.

[199] H. Zhu, L. Wang, C. Wang, Privacy-enhanced multi-user quantum private data query using partial quantum homomorphic encryption, Int. J. Theor. Phys. 60 (6)(2021) 2090-2101.