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New quantum key agreement protocols based on cluster states

  • Yu-Guang YangEmail author
  • Bo-Ran Li
  • Shuang-Yong Kang
  • Xiu-Bo Chen
  • Yi-Hua Zhou
  • Wei-Min Shi
Article
  • 24 Downloads

Abstract

A new two-party quantum key agreement (QKA) protocol is proposed based on four-qubit cluster states. Encoded four-qubit cluster states can be transmitted directly by means of order rearrangement operation. In contrast to existing QKA protocols based on four-qubit cluster states, it is unnecessary to perform two-way quantum communication. We analyze the security of this protocol and prove that it is secure in ideal conditions. We also propose the method to ensure the security of this protocol in noisy channel. Finally, we analyze the expansibility of the proposed QKA protocol and propose a three-party QKA protocol based on four-qubit cluster states.

Keywords

Quantum cryptography Quantum key agreement Cluster state 

Notes

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); and Jiangsu Postdoctoral Science Foundation.

References

  1. 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. IEEE, pp. 175–179 New York (1984)Google Scholar
  2. 2.
    Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–664 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89, 187902 (2002)ADSCrossRefGoogle Scholar
  5. 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)ADSCrossRefGoogle Scholar
  6. 6.
    Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59, 162–168 (1999)ADSCrossRefGoogle Scholar
  8. 8.
    Dušek, M., Haderka, O., Hendrych, M., Myska, R.: Quantum identification system. Phys. Rev. A 60, 149–156 (1999)ADSCrossRefGoogle Scholar
  9. 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Yang, Y.G., Cao, W.F., Wen, Q.Y.: Secure quantum private comparison. Phys. Scr. 80(6), 065002 (2009)ADSzbMATHCrossRefGoogle Scholar
  11. 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)ADSCrossRefGoogle Scholar
  12. 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)zbMATHCrossRefGoogle Scholar
  13. 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 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 CrossRefGoogle Scholar
  15. 15.
    Wang, T.-Y., Cai, X.Q., Ren, Y.L., Zhang, R.L.: Security of quantum digital signature. Sci. Rep. 5, 9231 (2015)CrossRefGoogle Scholar
  16. 16.
    Gao, F., Liu, B., Huang, W., Wen, Q.Y.: Postprocessing of the oblivious key in quantum private query. IEEE. J. Sel. Top. Quantum 21, 6600111 (2015)Google Scholar
  17. 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)ADSCrossRefGoogle Scholar
  18. 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. CHINA Phys. Mech. Astron. 60(12), 120311 (2017)Google Scholar
  19. 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)ADSzbMATHCrossRefGoogle Scholar
  20. 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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gao, F., Liu, B., Wen, Q.-Y.: Flexible quantum private queries based on quantum key distribution. Opt. Exp. 20, 17411–17420 (2012)ADSCrossRefGoogle Scholar
  22. 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)ADSMathSciNetCrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 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)CrossRefGoogle Scholar
  25. 25.
    Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory 22, 644–654 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Zhou, N., Zeng, G., Xiong, J.: Quantum key agreement protocol. Electron. Lett. 40(18), 1149 (2004)CrossRefGoogle Scholar
  27. 27.
    Tsai, C.W., Hwang, T.: On quantum key agreement protocol, Technical Report, CS-I-E, NCKU, Taiwan, ROC (2009)Google Scholar
  28. 28.
    Chong, S.K., Hwang, T.: Quantum key agreement protocol based on BB84. Opt. Commun. 283, 1192–1195 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    Shen, D.-S., Ma, W.-P., Wang, L.-L.: Two-party quantum key agreement with four-qubit cluster states. Quantum Inf. Process. 13, 2313–2324 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Sharma, V.: Effect of noise on practical quantum communication systems. Def. Sci. J. 66(2), 186–192 (2016)CrossRefGoogle Scholar
  31. 31.
    Sharma, R.D., Thapliyal, K., Pathak, A., et al.: Which verification qubits perform best for secure communication in noisy channel? Quantum Inf. Process. 15, 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Sharma, V., Thapliyal, K., Pathak, A., et al.: A comparative study of protocols for secure quantum communication under noisy environment: single-qubit-based protocols versus entangled-state-based protocols. Quantum Inf. Process. 15(11), 1–30 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    He, Y.-F., Ma, W.-P.: Two quantum key agreement protocols immune to collective noise. Int. J. Theor. Phys. 56, 328–338 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    He, Y.-F., Ma, W.-P.: Two-party quantum key agreement against collective noise. Quantum Inf. Process. 15, 5023–5035 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Shi, R.H., Zhong, H.: Multi-party quantum key agreement with Bell states and Bell measurements. Quantum Inf. Process. 12, 921–932 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Liu, B., Gao, F., Huang, W., Wen, Q.-Y.: Multiparty quantum key agreement with single particles. Quantum Inf. Process. 12, 1797–1805 (2013)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Yin, X.R., Ma, W.P., Liu, W.Y.: Three-party quantum key agreement with two-photon entanglement. Int. J. Theor. Phys. 52, 3915–3921 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    He, Y.-F., Ma, W.-P.: Quantum key agreement protocols with four-qubit cluster states. Quantum Inf. Process. 14, 3483–3498 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86(5), 910–913 (2001)ADSCrossRefGoogle Scholar
  42. 42.
    Raussendorf, R., Briegel, H.J.: A one-way quantum computer. Phys. Rev. Lett. 86, 5188–5191 (2001)ADSCrossRefGoogle Scholar
  43. 43.
    Shukla, C., Kothari, V., Banerjee, A., Pathak, A.: On the group-theoretic structure of a class of quantum dialogue protocols. Phys. Lett. A 377, 518–527 (2013)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Deng, F.-G., Long, G.L.: Controlled order rearrangement encryption for quantum key distribution. Phys. Rev. A 68, 042315 (2003)ADSCrossRefGoogle Scholar
  45. 45.
    Deng, F.-G., Long, G.L.: Quantum privacy amplification for a sequence of single qubits. Commun. Theor. Phys. 46, 443–446 (2006)ADSCrossRefGoogle Scholar
  46. 46.
    Hu, J., Yu, B., Jing, M., et al.: Experimental quantum secure direct communication with single photons. Light: Sci. Appl. 5, e16144 (2016)CrossRefGoogle Scholar
  47. 47.
    Qi, R.Y., Sun, Z., Lin, Z.S., et al.: Implementation and security analysis of practical quantum secure direct communication. arXiv:1810.11806
  48. 48.
    Lin, S., Wen, Q.-Y., Gao, F., Zhu, F.-C.: Quantum secure direct communication with χ-type entangled states. Phys. Rev. A 78, 064304 (2008)ADSCrossRefGoogle Scholar
  49. 49.
    Rivest, R.L.: Fast Software Encryption 97, LNCS 1267. Springer, Berlin (1997)Google Scholar
  50. 50.
    Cabello, A.: Quantum key distribution in the Holevo limit. Phys. Rev. Lett. 85, 5635–5638 (2000)ADSCrossRefGoogle Scholar
  51. 51.
    Gisin, N., Fasel, S., Kraus, B., Zbinden, H., Ribordy, G.: Trojan-horse attacks on quantum-key distribution systems. Phys. Rev. A 73, 022320 (2006)ADSCrossRefGoogle Scholar
  52. 52.
    Cai, Q.Y.: Eavesdropping on the two-way quantum communication protocols with invisible photons. Phys. Lett. A 351, 23–25 (2006)ADSzbMATHCrossRefGoogle Scholar
  53. 53.
    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)ADSzbMATHCrossRefGoogle Scholar
  54. 54.
    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)ADSCrossRefGoogle Scholar
  55. 55.
    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
  56. 56.
    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)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Gao, H., Chen, X.G., Qian, S.R.: Two-party quantum key agreement protocols under collective noise channel. Quantum Inf. Process. 17, 140 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    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
  59. 59.
    Wang, Y.H.: Beyond regular semigroups. Semigr. Forum 92(2), 414–448 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    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)Google Scholar
  61. 61.
    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
  62. 62.
    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
  63. 63.
    Cui, Y.J.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    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
  65. 65.
    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
  66. 66.
    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
  67. 67.
    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
  68. 68.
    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
  69. 69.
    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
  70. 70.
    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
  71. 71.
    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)MathSciNetCrossRefGoogle Scholar
  72. 72.
    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
  73. 73.
    Liu, F., Wu, H.X.: L-p bounds for Marcinkiewicz integrals associated to homogeneous mappings. Monatshefte Fur Math. 181(4), 875–906 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    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
  75. 75.
    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
  76. 76.
    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
  77. 77.
    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
  78. 78.
    Wang, Y.H.: Hall-type representations for generalised orthogroups. Semigroup Forum 89(3), 518–545 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    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
  80. 80.
    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
  81. 81.
    Ling, S.T., Cheng, X.H., Jiang, T.S.: An algorithm for coneigenvalues and coneigenvectors of quaternion matrices. Adv. Appl. Clifford Algebr. 25(2), 377–384 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    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
  83. 83.
    Liu, F., Wu, H.X.: On the regularity of the multisublinear maximal functions. Can. Math. Bul 58(4), 808–817 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    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
  85. 85.
    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)MathSciNetCrossRefGoogle Scholar
  86. 86.
    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
  87. 87.
    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
  88. 88.
    Zhang, Y.Q., Shen, D.M.: Estimation of semi-parametric varying-coefficient spatial panel data models with random-effects. J. Stat. Plan. Inference 159, 64–80 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    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
  90. 90.
    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
  91. 91.
    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
  92. 92.
    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
  93. 93.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  94. 94.
    Tan, C., Zhang, W.H.: On observability and detectability of continuous-time stochastic Markov jump systems. J. Syst. Sci. Complexity 28(4), 830–847 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Yan, Z.G., Zhang, G.S., Wang, J.K., Zhang, W.H.: State and output feedback finite-time guaranteed cost control of linear it stochastic systems. J. Syst. Sci. Complex. 28(4), 813–829 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Meng, X.Z., Zhao, S.N., Zhang, W.Y.: Adaptive dynamics analysis of a predator-prey model with selective disturbance. Appl. Math. Comput. 266, 946–958 (2015)MathSciNetGoogle Scholar
  97. 97.
    Cui, Y.J., Zou, Y.M.: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput. 256, 438–444 (2015)MathSciNetzbMATHGoogle Scholar
  98. 98.
    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
  99. 99.
    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
  100. 100.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    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
  102. 102.
    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
  103. 103.
    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
  104. 104.
    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. Mathematicas 112(2), 593–614 (2018)ADSGoogle Scholar
  105. 105.
    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
  106. 106.
    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
  107. 107.
    Cui, Y.J., Ma, W.J., Sun, Q., Su, X.W.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear. Analysis-Modelling Control. 23(1), 31–39 (2018)MathSciNetCrossRefGoogle Scholar
  108. 108.
    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. Eqn. 9, 1–10 (2018)zbMATHGoogle Scholar
  109. 109.
    Ma, W.-X.: Conservation laws by symmetries and adjoint symmetries. Discrete Continuous. Dyn. Systems-Series S 11(4), 707–721 (2018)MathSciNetzbMATHGoogle Scholar
  110. 110.
    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
  111. 111.
    Ma, W.-X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Eqn. 264(4), 2633–2659 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    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
  113. 113.
    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
  114. 114.
    Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequalities Appl. 21(1), 25–40 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    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
  116. 116.
    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
  117. 117.
    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
  118. 118.
    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
  119. 119.
    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. Bifurcat Chaos 27(13), 1750209 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    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
  121. 121.
    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
  122. 122.
    Zhang, R.Y., Xu, F.F., Huang, J.C.: Reconstructing local volatility using total variation. Acta Math. Sinica Engl. 33(2), 263–277 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  123. 123.
    Liu, F.: A remark on the regularity of the discrete maximal operator. Bull. Aust. Math. Soc. 95, 108–120 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Liu, F.: Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces. Math. Nachr. 290, 75–96 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  125. 125.
    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
  126. 126.
    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
  127. 127.
    Liu, F., Wu, H.X.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453, 144–158 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  128. 128.
    Liu, F., Wu, H.X.: Regularity of discrete multisublinear fractional maximal functions. Sci. China Math. 60(8), 1461–1476 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  129. 129.
    Liu, F., Wu, H.X.: Endpoint regularity of multisublinear fractional maximal functions. Can. Math. Bull. 60(3), 586–603 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  130. 130.
    Liu, F., Mao, S.Z.: On the regularity of the one-sided Hardy-Littlewood maximal functions. Czech. Math. J. 67(142), 219–234 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Liu, F.: On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math. Inequal. Appl. 20(2), 515–535 (2017)MathSciNetzbMATHGoogle Scholar
  132. 132.
    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
  133. 133.
    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
  134. 134.
    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
  135. 135.
    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
  136. 136.
    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
  137. 137.
    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
  138. 138.
    Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Compu. Math. Appl. 74, 591–596 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  139. 139.
    Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Compu. Math. Appl. 74, 1399–1405 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  140. 140.
    Liu, F., Wu, H.X.: Singular integrals related to homogeneous mappings in Triebel-Lizorkin spaces. J Math Inequalities 11(4), 1075–1097 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  141. 141.
    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

Copyright information

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

Authors and Affiliations

  • Yu-Guang Yang
    • 1
    Email author
  • Bo-Ran Li
    • 1
  • Shuang-Yong Kang
    • 2
  • Xiu-Bo Chen
    • 3
  • Yi-Hua Zhou
    • 1
  • Wei-Min Shi
    • 1
  1. 1.Faculty of Information TechnologyBeijing University of TechnologyBeijingChina
  2. 2.Institute of Information Applied TechnologyBeijingChina
  3. 3.Information Security Center, State Key Laboratory of Networking and Switching TechnologyBeijing University of Posts and TelecommunicationsBeijingChina

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