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Two-party quantum key agreement over a collective noisy channel

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Abstract

Quantum key agreement (QKA) allows participants to establish a shared key over a quantum channel, and no one of the participants can determine the shared key alone. Actually, particles are usually affected by noise during transmission in the quantum channel, and an aggressor can launch a baleful attack under the cover of noise. In this paper, based on logical Bell states, we propose two robust two-party QKA protocols immune to collective-dephasing noise and collective-rotation noise, respectively. The measurement correlation of quantum entanglement is utilized to establish a shared key. The proposed protocols are globally better in terms of quantum resource cost and qubit efficiency than existing two-party QKA protocols against collective noise. The security analysis demonstrates that they can resist common insider and outsider attacks.

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References

  1. Bennett, C. H., Brassard, G.: Quantum cryptography: public-key distribution and coin tossing. In: Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, pp. 175–179. IEEE, New York (1984)

  2. Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–664 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992)

    ADS  MathSciNet  MATH  Google Scholar 

  4. Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89, 187902 (2002)

    ADS  Google Scholar 

  5. Deng, F.G., Long, G.L., Liu, X.S.: Two-step quantum direct communication protocol using the Einstein–Podolsky–Rosen pair block. Phys. Rev. A 68, 042317 (2003)

    ADS  Google Scholar 

  6. Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59, 162–168 (1999)

    ADS  Google Scholar 

  8. Dušek, M., Haderka, O., Hendrych, M., Myska, R.: Quantum identification system. Phys. Rev. A 60, 149–156 (1999)

    ADS  Google Scholar 

  9. Yang, Y.G., Wen, Q.Y.: An efficient two-party quantum private comparison protocol with decoy photons and two-photon entanglement. J. Phys. A: Math. Theor. 42(5), 055305 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Yang, Y.G., Cao, W.F., Wen, Q.Y.: Secure quantum private comparison. Phys. Scr. 80(6), 065002 (2009)

    ADS  MATH  Google Scholar 

  11. Chen, X.B., Xu, G., Niu, X.X., Wen, Q.Y., Yang, Y.X.: An efficient protocol for the private comparison of equal information based on the triplet entangled state and single particle measurement. Opt. Commun. 283(7), 1561–1565 (2010)

    ADS  Google Scholar 

  12. Yang, Y.-G., Liu, Z.-C., Li, J., Chen, X.-B., Zuo, H.-J., Zhou, Y.-H., Shi, W.-M.: Theoretically extensible quantum digital signature with starlike cluster states. Quantum Inf. Process. 16(1), 1–15 (2017)

    MATH  Google Scholar 

  13. Yang, Y.-G., Lei, H., Liu, Z.-C., Zhou, Y.-H., Shi, W.-M.: Arbitrated quantum signature scheme based on cluster states. Quantum Inf. Process. 15(6), 2487–2497 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  14. Jiang, D.-H., Xu, Y.-L., Xu, G.-B.: Arbitrary quantum signature based on local indistinguishability of orthogonal product states. Int. J. Theor. Phys. (2019). https://doi.org/10.1007/s10773-018-03995-4

    Article  MathSciNet  Google Scholar 

  15. Wang, T.-Y., Cai, X.Q., Ren, Y.L., Zhang, R.L.: Security of quantum digital signature. Sci. Rep. 5, 9231 (2015)

    Google Scholar 

  16. Gao, F., Liu, B., Huang, W., Wen, Q.Y.: Postprocessing of the oblivious key in quantum private query. IEEE. J. Sel. Top. Quant. 21, 6600111 (2015)

    Google Scholar 

  17. Wei, C.Y., Wang, T.Y., Gao, F.: Practical quantum private query with better performance in resisting joint-measurement attack. Phys. Rev. A 93, 042318 (2016)

    ADS  Google Scholar 

  18. Yang, Y.-G., Liu, Z.-C., Chen, X.-B., Zhou, Y.-H., Shi, W.-M.: Robust QKD-based private database queries based on alternative sequences of single-qubit measurements. Sci. Chin. Phys. Mech. Astron. 60(12), 120311 (2017)

    ADS  Google Scholar 

  19. Yang, Y.-G., Liu, Z.-C., Li, J., Chen, X.-B., Zuo, H.-J., Zhou, Y.-H., Shi, W.-M.: Quantum private query with perfect user privacy against a joint-measurement attack. Phys. Lett. A 380(48), 4033–4038 (2016)

    ADS  MATH  Google Scholar 

  20. Yang, Y.-G., Liu, Z.C., Chen, X.B., Cao, W.F., Zhou, Y.H., Shi, W.M.: Novel classical post-processing for quantum key distribution-based quantum private query. Quantum Inf. Process. 15, 3833–3840 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Gao, F., Liu, B., Wen, Q.-Y.: Flexible quantum private queries based on quantum key distribution. Opt. Exp. 20, 17411–17420 (2012)

    ADS  Google Scholar 

  22. Yang, Y.-G., Sun, S.-J., Xu, P., Tian, J.: Flexible protocol for quantum private query based on B92 protocol. Quantum Inf. Process. 13, 805–813 (2014)

    ADS  MathSciNet  Google Scholar 

  23. Gao, F., Qin, S.J., Huang, W., Wen, Q.Y.: Quantum private query: a new kind of practical quantum cryptographic protocols. Sci. China-Phys. Mech. Astron. 62, 070301 (2019)

    Google Scholar 

  24. Yang, Y.-G., Guo, X.-P., Xu, G., Chen, X.-B., Li, J., Zhou, Y.-H., Shi, W.-M.: Reducing the communication complexity of quantum private database queries by subtle classical post-processing with relaxed quantum ability. Comput. Secur. 81, 15–24 (2019)

    Google Scholar 

  25. Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory 22, 644–654 (1976)

    MathSciNet  MATH  Google Scholar 

  26. Zhou, N., Zeng, G., Xiong, J.: Quantum key agreement protocol. Electron. Lett. 40(18), 1149 (2004)

    Google Scholar 

  27. Tsai, C.W., Hwang, T.: On quantum key agreement protocol, Technical Report, CS-I-E, NCKU, Taiwan, ROC, 2009

  28. Chong, S.K., Hwang, T.: Quantum key agreement protocol based on BB84. Opt. Commun. 283, 1192–1195 (2010)

    ADS  Google Scholar 

  29. He, Y.F., Ma, W.P.: Two-party quantum key agreement with five-particle entangled states. Int. J. Quantum Inf. 15(03), 1750018 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Cai, B., Guo, G., Lin, S.: Multi-party quantum key agreement with teleportation. Mod. Phys. Lett. B 31(10), 1750102 (2017)

    ADS  MathSciNet  Google Scholar 

  31. Cao, H., Ma, W.: Multiparty quantum key agreement based on quantum search algorithm. Sci. Rep. 7, 45046 (2017)

    ADS  Google Scholar 

  32. Shi, R.H., Zhong, H.: Multi-party quantum key agreement with Bell states and Bell measurements. Quantum Inf. Process. 12, 921–932 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  33. Liu, B., Gao, F., Huang, W., Wen, Q.-Y.: Multiparty quantum key agreement with single particles. Quantum Inf. Process. 12, 1797–1805 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  34. Xu, G.-B., Wen, Q.-Y., Gao, F., Qin, S.-J.: Novel multiparty quantum key agreement protocol with GHZ states. Quantum Inf. Process. 13, 2587–2594 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  35. Shukla, C., Alam, N., Pathak, A.: Protocols of quantum key agreement solely using Bell states and Bell measurement. Quantum Inf. Process. 13, 2391–2405 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  36. Sun, Z.W., Yu, J.P., Wang, P.: Efficient multi-party quantum key agreement by cluster States. Quantum Inf. Process. 15, 373–384 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  37. He, Y.-F., Ma, W.-P.: Quantum key agreement protocols with four-qubit cluster states. Quantum Inf. Process. 14, 3483–3498 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  38. Sun, Z.W., Zhang, C., Wang, P., Yu, J.P., Zhang, Y., Long, D.Y.: Multi-party quantum key agreement by an entangled six-qubit state. Int. J. Theor. Phys. 55, 1920–1929 (2016)

    MATH  Google Scholar 

  39. Huang, W., Wen, Q.-Y., Liu, B., Su, Q., Gao, F.: Cryptanalysis of a multi-party quantum key agreement protocol with single particles. Quantum Inf. Process. 13, 1651–1657 (2014)

    ADS  MathSciNet  Google Scholar 

  40. Huang, W., Su, Q., Xu, B.J., Liu, B., Fan, F., Jia, H.Y., Yang, Y.H.: Improved multiparty quantum key agreement in travelling mode. Sci. Chin. Phys. Mech. Astron. 59(12), 120311 (2016)

    Google Scholar 

  41. Sun, Z.W., Zhang, C., Wang, B.H., Li, Q., Long, D.Y.: Improvements on “multiparty quantum key agreement with single particles”. Quantum Inf. Process. 12, 3411–3420 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Kwiat, P.G., Berglund, A.J., Altepeter, J.B., White, A.G.: Experimental verification of decoherence-free subspaces. Science (New York N.Y.) 290(5491), 498–501 (2000)

    ADS  Google Scholar 

  43. Cai, B., Guo, G., Lin, S., Zuo, H., Yu, C.: Multipartite quantum key agreement over collective noise channels. IEEE Photonics J. 10(1), 1–11 (2018)

    Google Scholar 

  44. Huang, W., Wen, Q.Y., Liu, B., Gao, F., Sun, Y.: Quantum key agreement with EPR pairs and single particle measurements. Quantum Inf. Process. 13, 649–663 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  45. Huang, W., Su, Q., Wu, X., Li, Y.B., Sun, Y.: Quantum key agreement against collective decoherence. Int. J. Theor. Phys. 53, 2891 (2014)

    MATH  Google Scholar 

  46. He, Y.F., Ma, W.P.: Two-party quantum key agreement against collective noise. Quantum Inf. Process. 15, 5023–5035 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  47. He, Y.-F., Ma, W.-P.: Two quantum key agreement protocols immune to collective noise. Int. J. Theor. Phys. 56, 328–338 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Gao, H., Chen, X.G., Qian, S.R.: Two-party quantum key agreement protocols under collective noise channel. Quantum Inf. Process. 17, 140 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  49. Gisin, N., Fasel, S., Kraus, B., Zbinden, H., Ribordy, G.: Trojan-horse attacks on quantum-key distribution systems. Phys. Rev. A 73, 022320 (2006)

    ADS  Google Scholar 

  50. Cai, Q.Y.: Eavesdropping on the two-way quantum communication protocols with invisible photons. Phys. Lett. A 351, 23–25 (2006)

    ADS  MATH  Google Scholar 

  51. Vakhitov, A., Makarov, V., Hjelme, D.R.: Large pulse attack as a method of conventional optical eavesdropping in quantum cryptography. J. Mod. Opt. 48, 2023–2038 (2001)

    ADS  MATH  Google Scholar 

  52. Deng, F.G., Li, X.H., Zhou, H.Y., Zhang, Z.J.: Improving the security of multiparty quantum secret sharing against Trojan horse attack. Phys. Rev. A 72, 044302 (2005)

    ADS  Google Scholar 

  53. Deng, F.G., Zhou, P., Li, X.H., et al.: Robustness of two-way quantum communication protocols against Trojan horse attack. arXiv:quant-ph/0508168

  54. Gao, F., Qin, S.J., Wen, Q.Y., Zhu, F.C.: A simple participant attack on the Brádler–Dušek protocol. Quantum Inf. Comput. 7, 329–334 (2007)

    MathSciNet  MATH  Google Scholar 

  55. Gao, F., Guo, F.Z., Wen, Q.Y., Zhu, F.C.: Comment on “Experimental demonstration of a quantum protocol for byzantine agreement and liar detectio”. Phys. Rev. Lett. 101, 208901 (2008)

    ADS  Google Scholar 

  56. Qin, S., Gao, F., Wen, Q., Zhu, F.: Improving the security of multiparty quantum secret sharing against an attack with a fake signal. Phys. Lett. A 357, 101–103 (2006)

    ADS  MATH  Google Scholar 

  57. Cabello, A.: Quantum key distribution in the Holevo limit. Phys. Rev. Lett. 85, 5635–5638 (2000)

    ADS  Google Scholar 

  58. Jiang, D.-H., Wang, X.-J., Xu, G.-B., Lin, J.-Q.: A denoising-decomposition model combining TV minimisation and fractional derivatives. East Asia J. Appl. Math. 8, 447–462 (2018)

    Google Scholar 

  59. Li, L., Wang, Z., Li, Y.X., Shen, H., Lu, J.W.: Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays. Appl. Math. Comput. 330, 152–169 (2018)

    MathSciNet  Google Scholar 

  60. Liang, X., Gao, F., Zhou, C.-B., Wang, Z., Yang, X.-J.: An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type. Adv. Differ. Eqn. 2018, 25 (2018)

    MathSciNet  MATH  Google Scholar 

  61. Wang, J., Liang, K., Huang, X., Wang, Z., Shen, H.: Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback. Appl. Math. Comput. 328, 247–262 (2018)

    MathSciNet  Google Scholar 

  62. Zhou, J.P., Sang, C.Y., Li, X., Fang, M.Y., Wang, Z.: H∞ consensus for nonlinear stochastic multi-agent systems with time delay. Appl. Math. Comput. 325, 41–58 (2018)

    MathSciNet  Google Scholar 

  63. Hu, Q.Y., Yuan, L.: A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. Adv. Comput. Math. 44(1), 245–275 (2018)

    MathSciNet  MATH  Google Scholar 

  64. Liu, F.: Rough maximal functions supported by subvarieties on Triebel-Lizorkin spaces, Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Serie A. Math. 112(2), 593–614 (2018)

    Google Scholar 

  65. Wang, W., Zhang, T.Q.: Caspase-1-mediated pyroptosis of the predominance for driving CD4++ T cells death: a nonlocal spatial mathematical model. Bull. Math. Biol. 80(3), 540–582 (2018)

    MathSciNet  MATH  Google Scholar 

  66. Li, H.J., Zhu, Y.L., Liu, J., Wang, Y.: Consensus of second-order delayed nonlinear multi-agent systems via node-based distributed adaptive completely intermittent protocols. Appl. Math. Comput. 326, 1–15 (2018)

    MathSciNet  Google Scholar 

  67. Cui, Y.J., Ma, W.J., Sun, Q., Su, X.W.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal. Model. Control 23(1), 31–39 (2018)

    MathSciNet  Google Scholar 

  68. Cui, Y.J., Ma, W.J., Wang, X.Z., Su, X.W.: Uniqueness theorem of differential system with coupled integral boundary conditions. Electron. J. Qual. Theory Differ. Equ. 9, 1–10 (2018)

    MathSciNet  MATH  Google Scholar 

  69. Ma, W.-X.: Conservation laws by symmetries and adjoint symmetries. Discrete. Contin. Dyn. Syst. Ser. S 11(4), 707–721 (2018)

    MathSciNet  MATH  Google Scholar 

  70. Ma, W.-X., Yong, X.L., Zhang, H.-Q.: Diversity of interaction solutions to the (2+1)-dimensional Ito equation. Comput. Math. Appl. 75(1), 289–295 (2018)

    MathSciNet  Google Scholar 

  71. Ma, W.-X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)

    ADS  MathSciNet  MATH  Google Scholar 

  72. McAnally, M., Ma, W.-X.: An integrable generalization of the D-Kaup–Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy. Appl. Math. Comput. 323, 220–227 (2018)

    MathSciNet  Google Scholar 

  73. Lu, C.N., Fu, C., Yang, H.W.: Time-fractional generalized boussinesq equation for rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions. Appl. Math. Comput. 327, 104–116 (2018)

    MathSciNet  Google Scholar 

  74. Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018)

    MathSciNet  MATH  Google Scholar 

  75. Wang, J., Cheng, H., Li, Y., et al.: The geometrical analysis of a predator-prey model with multi-state dependent impulsive. J. Appl. Anal. Comput. 8(2), 427–442 (2018)

    MathSciNet  Google Scholar 

  76. Chen, J., Zhang, T., Zhang, Z.Y., Lin, C., Chen, B.: Stability and output feedback control for singular Markovian jump delayed systems. Math. Control Relat. Fields 8(2), 475–490 (2018)

    MathSciNet  Google Scholar 

  77. Xu, X.-X., Sun, Y.-P.: Two symmetry constraints for a generalized Dirac integrable hierarchy. J. Math. Anal. Appl. 458, 1073–1090 (2018)

    MathSciNet  MATH  Google Scholar 

  78. Shen, H., Song, X.N., Li, F., Wang, Z., Chen, B.: Finite-time L2-L∞ filter design for networked Markov switched singular systems: a unified method. Appl. Math. Comput. 321(15), 450–462 (2018)

    MathSciNet  Google Scholar 

  79. Wang, Z., Wang, X.H., Li, Y.X., Huang, X.: Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay. Int. J. Bifurc. Chaos 27(13), 1750209 (2017)

    MathSciNet  MATH  Google Scholar 

  80. Zhang, Y., Dong, H.H., Zhang, X.E., Yang, H.W.: Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation. Comput. Math. Appl. 73, 246–252 (2017)

    MathSciNet  MATH  Google Scholar 

  81. Zhang, S.Q., Meng, X.Z., Zhang, T.H.: Dynamics analysis and numerical simulations of a stochastic non-autonomous predator-prey system with impulsive effects. Nonlinear Anal: Hybrid Syst. 26, 19–37 (2017)

    MathSciNet  MATH  Google Scholar 

  82. Zhang, R.Y., Xu, F.F., Huang, J.C.: Reconstructing local volatility using total variation. Acta Math. Sin.-English Ser. 33(2), 263–277 (2017)

    MathSciNet  MATH  Google Scholar 

  83. Liu, F.: A remark on the regularity of the discrete maximal operator. Bull. Aust. Math. Soc. 95, 108–120 (2017)

    MathSciNet  MATH  Google Scholar 

  84. Liu, F.: Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces. Math. Nachr. 290, 75–96 (2017)

    MathSciNet  MATH  Google Scholar 

  85. Tian, Z.L., Tian, M.Y., Liu, Z.Y., Xu, T.Y.: The Jacobi and Gauss-Seidel-type iteration methods for the matrix equation AXB = C. Appl. Math. Comput. 292, 63–75 (2017)

    MathSciNet  Google Scholar 

  86. Song, Q.L., Dong, X.Y., Bai, Z.B., Chen, B.: Existence for fractional Dirichlet boundary value problem under barrier strip conditions. J. Nonlinear Sci. Appl. 10, 3592–3598 (2017)

    MathSciNet  Google Scholar 

  87. Liu, F., Wu, H.X.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453, 144–158 (2017)

    MathSciNet  MATH  Google Scholar 

  88. Liu, F., Wu, H.X.: Regularity of discrete multisublinear fractional maximal functions. Sci. China Math. 60(8), 1461–1476 (2017)

    MathSciNet  MATH  Google Scholar 

  89. Liu, F., Wu, H.X.: Endpoint regularity of multisublinear fractional maximal functions. Can. Math. Bull. 60(3), 586–603 (2017)

    MathSciNet  MATH  Google Scholar 

  90. Liu, F., Mao, S.Z.: On the regularity of the one-sided Hardy-Littlewood maximal functions. Czech. Math. J. 67(142), 219–234 (2017)

    MathSciNet  MATH  Google Scholar 

  91. Liu, F.: On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math. Inequal. Appl. 20(2), 515–535 (2017)

    MathSciNet  MATH  Google Scholar 

  92. Li, X.Y., Zhao, Q.L.: A new integrable symplectic map by the binary nonlinearization to the super AKNS system. J. Geom. Phys. 121, 123–137 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  93. Cheng, W., Xu, J.F., Cui, Y.J.: Positive solutions for a system of nonlinear semipositone fractional q-difference equations with q-integral boundary conditions. J. Nonlinear Sci. Appl. 10, 4430–4440 (2017)

    MathSciNet  Google Scholar 

  94. Xu, X.-X., Sun, Y.-P.: An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation. J. Nonlinear Sci. Appl. 10, 3328–3343 (2017)

    MathSciNet  Google Scholar 

  95. Liu, Y.Q., Sun, H.G., Yin, X.L., Xin, B.G.: A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations. J. Nonlinear Sci. Appl. 10, 4515–4523 (2017)

    MathSciNet  Google Scholar 

  96. Chen, J.C., Zhu, S.D.: Residual symmetries and soliton-cnoidal wave interaction solutions for the negative-order Korteweg-de Vries equation. Appl. Math. Lett. 73, 136–142 (2017)

    MathSciNet  MATH  Google Scholar 

  97. Zhang, X.E., Chen, Y., Zhang, Y.: Breather, lump and X soliton solutions to nonlocal KP equation. Comput. Math. Appl. 74(10), 2341–2347 (2017)

    MathSciNet  MATH  Google Scholar 

  98. Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)

    MathSciNet  MATH  Google Scholar 

  99. Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)

    MathSciNet  MATH  Google Scholar 

  100. Liu, F., Wu, H.X.: Singular integrals related to homogeneous mappings in Triebel-Lizorkin spaces. J. Math. Inequal. 11(4), 1075–1097 (2017)

    MathSciNet  MATH  Google Scholar 

  101. Liu, F.: Rough singular integrals associated to surfaces of revolution on Triebel-Lizorkin spaces. Rocky Mt. J. Math. 47(5), 1617–1653 (2017)

    MathSciNet  MATH  Google Scholar 

  102. Zhao, Q.L., Li, X.Y.: A bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy. Anal. Math. Phys. 6(3), 237–254 (2016)

    MathSciNet  MATH  Google Scholar 

  103. Wang, Y.H.: Beyond regular semigroups. Semigroup Forum 92(2), 414–448 (2016)

    MathSciNet  MATH  Google Scholar 

  104. Zhang, J.K., Wu, X.J., Xing, L.S., Zhang, C.: Bifurcation analysis of five-level cascaded H-bridge inverter using proportional-resonant plus time-delayed feedback. Int. J. Bifurc. Chaos 26(11), 1630031 (2016)

    MathSciNet  MATH  Google Scholar 

  105. Zhang, T.Q., Meng, X.Z., Zhang, T.H.: Global analysis for a delayed siv model with direct and environmental transmissions. J. Appl. Anal. Comput. 6(2), 479–491 (2016)

    MathSciNet  Google Scholar 

  106. Meng, X.Z., Wang, L., Zhang, T.H.: Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment. J. Appl. Anal. Comput. 6(3), 865–875 (2016)

    MathSciNet  Google Scholar 

  107. Cui, Y.J.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)

    MathSciNet  MATH  Google Scholar 

  108. Meng, X.Z., Zhao, S.N., Feng, T., Zhang, T.H.: Dynamics of a novel nonlinear stochastic Sis epidemic model with double epidemic hypothesis. J. Math. Anal. Appl. 433(1), 227–242 (2016)

    MathSciNet  MATH  Google Scholar 

  109. Yin, C., Cheng, Y.H., Zhong, S.M., Bai, Z.B.: Fractional-order switching type control law design for adaptive sliding mode technique of 3d fractional-order nonlinear systems. Complexity 21(6), 363–373 (2016)

    MathSciNet  Google Scholar 

  110. Liu, F., Mao, S.Z., Wu, H.X.: On rough singular integrals related to homogeneous mappings. Collectanea Math. 67(1), 113–132 (2016)

    MathSciNet  MATH  Google Scholar 

  111. Liu, F., Chen, T., Wu, H.X.: A note on the endpoint regularity of the Hardy-littlewood maximal functions. Bull. Aust. Math. Soc. 94(1), 121–130 (2016)

    MathSciNet  MATH  Google Scholar 

  112. Liu, F., Fu, Z.W., Zheng, Y.P., Yuan, Q.: A rough marcinkiewicz integral along smooth curves. J. Nonlinear Sci. Appl. 9(6), 4450–4464 (2016)

    MathSciNet  MATH  Google Scholar 

  113. Liu, F., Wang, F.: Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points. Acta Math. Sin.-English Ser. 32(4), 507–520 (2016)

    MathSciNet  MATH  Google Scholar 

  114. Cui, Y.J.: Existence of solutions for coupled integral boundary value problem at resonance. Publ. Mathe-Debrecen 89(1–2), 73–88 (2016)

    MathSciNet  MATH  Google Scholar 

  115. Cui, Y.J., Zou, Y.M.: Existence of solutions for second-order integral boundary value problems. Nonlinear Anal. Model. Control 21(6), 828–838 (2016)

    MathSciNet  Google Scholar 

  116. Dong, H.H., Guo, B.Y., Yin, B.S.: Generalized fractional supertrace identity for Hamiltonian structure of Nls-Mkdv hierarchy with self-consistent sources. Anal. Math. Phys. 6(2), 199–209 (2016)

    MathSciNet  MATH  Google Scholar 

  117. Liu, F., Wu, H.X.: L-p bounds for marcinkiewicz integrals associated to homogeneous mappings. Monatshefte für Mathematik 181(4), 875–906 (2016)

    MathSciNet  MATH  Google Scholar 

  118. Li, X.P., Lin, X.Y., Lin, Y.Q.: Lyapunov-Type conditions and stochastic differential equations driven by G-brownian motion. J. Math. Anal. Appl. 439(1), 235–255 (2016)

    MathSciNet  MATH  Google Scholar 

  119. Liu, F., Zhang, D.Q.: Multiple singular integrals and maximal operators with mixed homogeneity along compound surfaces. Math. Inequal. Appl. 19(2), 499–522 (2016)

    MathSciNet  MATH  Google Scholar 

  120. Zhao, Y., Zhang, W.H.: Observer-based controller design for singular stochastic Markov jump systems with state dependent noise. J. Syst. Sci. Complex. 29(4), 946–958 (2016)

    MathSciNet  MATH  Google Scholar 

  121. Ma, H.J., Jia, Y.M.: Stability analysis for stochastic differential equations with infinite Markovian switchings. J. Math. Anal. Appl. 435(1), 593–605 (2016)

    MathSciNet  MATH  Google Scholar 

  122. Zhang, T.Q., Ma, W.B., Meng, X.Z., Zhang, T.H.: Periodic solution of a prey-predator model with nonlinear state feedback control. Appl. Math. Comput. 266, 95–107 (2015)

    MathSciNet  Google Scholar 

  123. Liu, F., Zhang, D.Q.: Parabolic marcinkiewicz integrals associated to polynomials compound curves and extrapolation. Bull. Korean Math. Soc. 52(3), 771–788 (2015)

    MathSciNet  MATH  Google Scholar 

  124. Ling, S.T., Cheng, X.H., Jiang, T.S.: An algorithm for coneigenvalues and coneigenvectors of quaternion matrices. Adv. Appl. Clifford Algebras 25(2), 377–384 (2015)

    MathSciNet  MATH  Google Scholar 

  125. Liu, F., Wu, H.X., Zhang, D.Q.: L-p bounds for parametric marcinkiewicz integrals with mixed homogeneity. Math. Inequal. Appl. 18(2), 453–469 (2015)

    MathSciNet  MATH  Google Scholar 

  126. Liu, F., Wu, H.X.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58(4), 808–817 (2015)

    MathSciNet  MATH  Google Scholar 

  127. Gao, M., Sheng, L., Zhang, W.H.: Stochastic H-2/H-infinity control of nonlinear systems with time-delay and state-dependent noise. Appl. Math. Comput. 266, 429–440 (2015)

    MathSciNet  Google Scholar 

  128. Li, Y.X., Huang, X., Song, Y.W., Lin, J.N.: A new fourth-order memristive chaotic system and its generation. Int. J. Bifurc. Chaos 25(11), 1550151 (2015)

    MathSciNet  Google Scholar 

  129. Xu, X.X.: A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and darboux transformation. Appl. Math. Comput. 251, 275–283 (2015)

    MathSciNet  MATH  Google Scholar 

  130. Li, X.Y., Zhao, Q.L., Li, Y.X., Dong, H.H.: Binary bargmann symmetry constraint associated with 3×3 discrete matrix spectral problem. J. Nonlinear Sci. Appl. 8(5), 496–506 (2015)

    MathSciNet  MATH  Google Scholar 

  131. Yu, J., Li, M.Q., Wang, Y.L., He, G.P.: A decomposition method for large-scale box constrained optimization. Appl. Math. Comput. 231, 9–15 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61572053); Beijing Natural Science Foundation (Grant No. 4182006); the National Natural Science Foundation of China (Grant Nos. 61671087, U1636106, 61602019, 61571226, 61701229,61702367); Natural Science Foundation of Jiangsu Province, China (Grant No. BK20170802); Jiangsu postdoctoral science foundation.

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Authors and Affiliations

  1. Faculty of Information Technology, Beijing University of Technology, Beijing, 100124, China

    Yu-Guang Yang, Shang Gao, Yi-Hua Zhou & Wei-Min Shi

  2. College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Dan Li

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  1. Yu-Guang Yang
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  2. Shang Gao
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Correspondence to Yu-Guang Yang.

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Yang, YG., Gao, S., Li, D. et al. Two-party quantum key agreement over a collective noisy channel. Quantum Inf Process 18, 74 (2019). https://doi.org/10.1007/s11128-019-2187-8

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