Verifiable Quantum Secret Sharing Protocols Based on Four-Qubit Entangled States

  • Wei-Feng CaoEmail author
  • Yu-Guang Yang


A new verifiable three-party quantum secret sharing protocol is proposed based on a special four-qubit entangled state which is inequivalent to four-qubit singlet states under stochastic local operations and classical communication (SLOCC). The validity of the reconstructed secret is verifiable based on the method like Byzantine agreement. It is generalized to verifiable multiparty quantum secret sharing based on four-qubit entangled states. It is shown to be secure against common attacks and feasible with present-day technology. PACS number(s): 03.67.Dd, 03.67.Hk, 03.67.Lx.


Quantum cryptography Quantum secret sharing Four-qubit entangled state 



This work was supported by the National Natural Science Foundation of China (Grant Nos.61572053,61671087,61602019); Beijing Natural Science Foundation (Grant No. 4182006).


  1. 1.
    Shamir, A.: How to share a secret. Commun. ACM. 22, 612–613 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proceedings of National Computer Conference, pp. 313–317. AFIPS, New York (1979)Google Scholar
  3. 3.
    Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults. In: Proceedings of 26th IEEE Symposium on Foundations of Computer Science, pp. 383–395 (1985)Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–664 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89, 187902 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    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)ADSCrossRefGoogle Scholar
  9. 9.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A. 59, 1829–1834 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cleve, R., Gottesman, D., Lo, H.K.: How to share a quantum secret. Phys. Rev. Lett. 83(3), 648–651 (1999)ADSCrossRefGoogle Scholar
  11. 11.
    Dušek, M., Haderka, O., Hendrych, M., Myska, R.: Quantum identification system. Phys. Rev. A. 60, 149–156 (1999)ADSCrossRefGoogle Scholar
  12. 12.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Yang, Y.G., Cao, W.F., Wen, Q.Y.: Secure quantum private comparison. Phys. Scr. 80(6), 065002 (2009)ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    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)ADSCrossRefGoogle Scholar
  15. 15.
    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)zbMATHCrossRefGoogle Scholar
  16. 16.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    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).
  18. 18.
    Wang, T.-Y., Cai, X.Q., Ren, Y.L., Zhang, R.L.: Security of quantum digital signature. Sci. Rep. 5, 9231 (2015)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    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)ADSCrossRefGoogle Scholar
  21. 21.
    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. China Phys. Mech. Astron. 60(12), 120311 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    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)ADSzbMATHCrossRefGoogle Scholar
  23. 23.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Gao, F., Liu, B., Wen, Q.-Y.: Flexible quantum private queries based on quantum key distribution. Opt. Express. 20, 17411–17420 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    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)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, J.-L., Guo, F.-Z., Gao, F., Liu, B., Wen, Q.-Y.: Private database queries based on counterfactual quantum key distribution. Phys. Rev. A. 88, 022334 (2013)ADSCrossRefGoogle Scholar
  27. 27.
    Wei, C.Y., Cai, X.Q., Liu, B., Wang, T.Y., Gao, F.: A generic construction of quantum-oblivious-key-transfer-based private query with ideal database security and zero failure. IEEE Trans. Comput. 67, 2–8 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    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
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    Yang, Y.-G., Yang, J.-J., Zhou, Y.-H., Shi, W.-M., Chen, X.-B., Li, J., Zuo, H.-J.: Quantum network communication: a discrete-time quantum-walk approach. SCIENCE CHINA Inf. Sci. 61(4), 042501 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Xu, G., Chen, X.B., Zhao, D., Li, Z.P., Yang, Y.X.: A novel protocol for multiparty quantum key management. Quantum Inf. Process. 14, 2959–2980 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Xu, G., Chen, X.B., Li, J., Wang, C., Yang, Y.X., Li, Z.: Network coding for quantum cooperative multicast. Quantum Inf. Process. 14, 4297–4322 (2015)Google Scholar
  33. 33.
    Li, J., Chen, X.B., Xu, G., Yang, Y.X., Li, Z.P.: Perfect quantum network coding independent of classical network solutions. IEEE Commun. Lett. 19, 115–118 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    Gottesman, D.: Theory of quantum secret sharing. Phys. Rev. A. 61(4), 042311 (2000)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A. 59(1), 162–168 (1999)ADSCrossRefGoogle Scholar
  36. 36.
    Deng, F.G., Li, X.H., Li, C.Y., Zhou, P., Zhou, H.Y.: Multiparty quantum secret splitting and quantum state sharing. Phys. Lett. A. 354(3), 190–195 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Xiao, L., Long, G.L., Deng, F.G., Pan, J.W.: Efficient multiparty quantum-secret-sharing schemes. Phys. Rev. A. 69(5), 052307 (2004)ADSCrossRefGoogle Scholar
  38. 38.
    Ji, Q., Liu, Y., Xie, C., Yin, X., Zhang, Z.: Tripartite quantum operation sharing with two asymmetric three-qubit W states in five entanglement structures. Quantum Inf. Process. 13, 1659–1676 (2014)ADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Yang, Y.-G., Cao, W.-F., Wen, Q.-Y.: Three-party quantum secret sharing of secure direct communication based on χ-type entangled states. Chin. Phys. B. 19(5), 050306 (2010)ADSCrossRefGoogle Scholar
  40. 40.
    Dehkordi, M.H., Fattahi, E.: Threshold quantum secret sharing between multiparty and multiparty using Greenberger–Horne–Zeilinger state. Quantum Inf. Process. 12, 1299–1306 (2013)ADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Guo, Y., Zhao, Y.Q.: High-efficient quantum secret sharing based on the Chinese remainder theorem via the orbital angular momentum entanglement analysis. Quantum Inf. Process. 12, 1125–1139 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Liao, C., Yang, C., Hwang, T.: Dynamic quantum secret sharing protocol based on GHZ state. Quantum Inf. Process. 13(8), 1907–1916 (2014)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    He, X., Yang, C.: Deterministic transfer of multiqubit GHZ entangled states and quantum secret sharing between different cavities. Quantum Inf. Process. 14(12), 4461–4474 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Karimipour, V., Asoudeh, M.: Quantum secret sharing and random hopping: using single states instead of entanglement. Phys. Rev. A 92: 030301(R)(2015)Google Scholar
  45. 45.
    Yang, Y.G., Teng, Y.W., Chai, H.P., Wen, Q.Y.: Verifiable quantum (k, n)-threshold secret key sharing. Int. J. Theor. Phys. 50, 792–798 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Yang,Y.G., Jia, X.,Wang, H.Y., Zhang, H.: Verifiable quantum (k, n)-threshold secret sharing. Quantum Inf. Process 11: 1619–1625(2012)Google Scholar
  47. 47.
    Xu, J., Yuan, J.: Improvement and extension of quantum secret sharing using orthogonal product states. Int. J. Quantum Inf. 12(1), 1450008 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Markham, D., Sanders, B.C.: Graph states for quantum secret sharing. Phys. Rev. A. 78(4), 042309 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Keet, A., Forescue, B., Markham, D., Sanders, B.C.: Quantum secret sharing with qudit graph states. Phys. Rev. A. 82(6), 062315 (2010)ADSCrossRefGoogle Scholar
  50. 50.
    Sarvepalli, P.: Nonthreshold quantum secret-sharing schemes in the graph-state formalism. Phys. Rev. A. 86(4), 042303 (2012)ADSCrossRefGoogle Scholar
  51. 51.
    Tavakoli, A., Herbauts, I., Zukowski, M., Bourennane, M.: Secret sharing with a single d-level quantum system. Phys. Rev. A 92: 030302(R)(2015)Google Scholar
  52. 52.
    Lin, S., Guo, G., Xu, Y., Sun, Y., Liu, X.: Cryptanalysis of quantum secret sharing with d-level single particles. Phys. Rev. A. 93, 062343 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    Lau, H.K., Weedbrook, C.: Quantum secret sharing with continuous-variable cluster states. Phys. Rev. A. 88, 042313 (2013)ADSCrossRefGoogle Scholar
  54. 54.
    Li, D.F., Li, X.R., Huang, H.T., Li, X.X.: Classification of four-qubit states by means of a stochastic local operation and the classical communication invariant and semi-invariants. Phys. Rev. A. 76, 052311 (2007)ADSCrossRefGoogle Scholar
  55. 55.
    Zhang, Y.Q., Shen, D.M.: Estimation of semi-parametric varying-coefficient spatial panel data models with random-effects. J. Statist. Plann. Inference. 159, 64–80 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Dong, H.H., Zhao, K., Yang, H.W., Li, Y.Q.: Generalised (2+1)-dimensional super Mkdv hierarchy for integrable systems in soliton theory. East Asian J. Appl. Math. 5(3), 256–272 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Liu, F., Wang, Z.Y., Wang, F.: Hamiltonian systems with positive topological entropy and conjugate points. J. Appl. Anal. Comput. 5(3), 527–533 (2015)MathSciNetGoogle Scholar
  58. 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. 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)MathSciNetGoogle Scholar
  60. 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. Difference Equ. 2018(25), (2018)Google Scholar
  61. 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)MathSciNetGoogle Scholar
  62. 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)MathSciNetGoogle Scholar
  63. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Liu, F.: Rough maximal functions supported by subvarieties on Triebel-Lizorkin spaces, Revista de la Real Academia de Ciencias Exactas. Fisicas y Nat. Ser. A. Math. 112(2), 593–614 (2018)Google Scholar
  65. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 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)MathSciNetCrossRefGoogle Scholar
  67. 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. Modell. Control. 23(1), 31–39 (2018)MathSciNetCrossRefGoogle Scholar
  68. 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)Google Scholar
  69. 69.
    Ma, W.-X.: Conservation laws by symmetries and adjoint symmetries. Discrete Contin. Dynam. Systems-Series S. 11(4), 707–721 (2018)MathSciNetzbMATHGoogle Scholar
  70. 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)MathSciNetCrossRefGoogle Scholar
  71. 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  72. 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)MathSciNetGoogle Scholar
  73. 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)MathSciNetGoogle Scholar
  74. 74.
    Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018)MathSciNetzbMATHGoogle Scholar
  75. 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)MathSciNetGoogle Scholar
  76. 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)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Xu, X.-X., Sun, Y.-P.: Two symmetry constraints for a generalized Dirac integrable hierarchy. J. Math. Anal. Appl. 458, 1073–1090 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 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)MathSciNetGoogle Scholar
  79. 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. Bifurcation Chaos. 27(13), 1750209 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  80. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Zhang, R.Y., Xu, F.F., Huang, J.C.: Reconstructing Local Volatility Using Total Variation. Acta Math. Sin. Engl. 33(2), 263–277 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Liu, F.: A remark on the regularity of the discrete maximal operator. Bull. Aust. Math. Soc. 95, 108–120 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Liu, F.: Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces. Math. Nachr. 290, 75–96 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 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)MathSciNetGoogle Scholar
  86. 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)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Liu, F., Wu, H.X.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453, 144–158 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Liu, F., Wu, H.X.: Regularity of discrete multisublinear fractional maximal functions. SCIENCE CHINA Math. 60(8), 1461–1476 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    Liu, F., Wu, H.X.: Endpoint regularity of multisublinear fractional maximal functions. Can. Math. Bull. 60(3), 586–603 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Liu, F., Mao, S.Z.: On the regularity of the one-sided Hardy-Littlewood maximal functions. Czechoslov. Math. J. 67(142), 219–234 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Liu, F.: On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math. Inequal. Appl. 20(2), 515–535 (2017)MathSciNetzbMATHGoogle Scholar
  92. 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  93. 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)MathSciNetCrossRefGoogle Scholar
  94. 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)MathSciNetCrossRefGoogle Scholar
  95. 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)MathSciNetCrossRefGoogle Scholar
  96. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Liu, F., Wu, H.X.: Singular integrals related to homogeneous mappings in Triebel-Lizorkin spaces. J. Math. Inequal. 11(4), 1075–1097 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Liu, F.: Rough singular integrals associated to surfaces of revolution on Triebel-Lizorkin spaces. Rocky Mt. J. Math. 47(5), 1617–1653 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Wang, Y.H.: Beyond regular semigroups. Semigroup Forum. 92(2), 414–448 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 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. Bifurcation Chaos. 26(11), 1630031 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  105. 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)MathSciNetGoogle Scholar
  106. 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)MathSciNetGoogle Scholar
  107. 107.
    Cui, Y.J.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 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)MathSciNetCrossRefGoogle Scholar
  110. 110.
    Liu, F., Mao, S.Z., Wu, H.X.: On rough singular integrals related to homogeneous mappings. Collect. Math. 67(1), 113–132 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Liu, F., Wang, F.: Entropy-expansiveness of geodesic flows on closed manifolds without conjugate points. Acta Math. Sin. (Engl. Ser.). 32(4), 507–520 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Cui, Y.J.: Existence of solutions for coupled integral boundary value problem at resonance. Publ. Math. Debr. 89(1-2), 73–88 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Cui, Y.J., Zou, Y.M.: Existence of solutions for second-order integral boundary value problems. Nonlinear Anal. Modell. Control. 21(6), 828–838 (2016)MathSciNetCrossRefGoogle Scholar
  116. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Liu, F.: Wu, H.X.: L-p bounds for marcinkiewicz integrals associated to homogeneous mappings. Monatsh. Math. 181(4), 875–906 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  118. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 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)MathSciNetzbMATHGoogle Scholar
  120. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  121. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 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)MathSciNetGoogle Scholar
  123. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Ling, S.T., Cheng, X.H., Jiang, T.S.: An algorithm for coneigenvalues and coneigenvectors of quaternion matrices. AACA. 25(2), 377–384 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 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)MathSciNetzbMATHGoogle Scholar
  126. 126.
    Liu, F., Wu, H.X.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. - Bulletin Canadien De Mathematiques. 58(4), 808–817 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  127. 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)MathSciNetGoogle Scholar
  128. 128.
    Li, Y.X., Huang, X., Song, Y.W., Lin, J.N.: A new fourth-order memristive chaotic system and its generation. Int. J. Bifurcation Chaos. 25(11), 1550151 (2015)ADSMathSciNetCrossRefGoogle Scholar
  129. 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)MathSciNetzbMATHGoogle Scholar
  130. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  131. 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)MathSciNetzbMATHGoogle Scholar
  132. 132.
    Liu, F., Mao, S.Z.: L-p bounds for nonisotropic marcinkiewicz integrals associated to surfaces. J. Aust. Math. Soc. 99(3), 380–398 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Tramontana, F., Elsadany, A.A., Xin, B.G., Agiza, H.N.: Local stability of the cournot solution with increasing heterogeneous competitors. Nonlinear Anal. Real World Appl. 26, 150–160 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  134. 134.
    Cui, Y.J., Zou, Y.M.: Monotone iterative technique for (K,N-K) conjugate boundary value problems. Electron. J. Qual. Theory Differ. Equ. (69), 1–11 (2015)Google Scholar
  135. 135.
    Tan, C., Zhang, W.H.: On observability and detectability of continuous-time stochastic Markov jump systems. J. Syst. Sci. Complex. 28(4), 830–847 (2015)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Electric and Information EngineeringZhengzhou University of Light IndustryZhengzhouChina
  2. 2.Faculty of Information TechnologyBeijing University of TechnologyBeijingChina

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