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.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
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)
Ekert, A.: Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–664 (1991)
Bennett, C.H.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 68, 3121–3124 (1992)
Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89, 187902 (2002)
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)
Hillery, M., Bužek, V., Berthiaume, A.: Quantum secret sharing. Phys. Rev. A 59, 1829–1834 (1999)
Karlsson, A., Koashi, M., Imoto, N.: Quantum entanglement for secret sharing and secret splitting. Phys. Rev. A 59, 162–168 (1999)
Dušek, M., Haderka, O., Hendrych, M., Myska, R.: Quantum identification system. Phys. Rev. A 60, 149–156 (1999)
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)
Yang, Y.G., Cao, W.F., Wen, Q.Y.: Secure quantum private comparison. Phys. Scr. 80(6), 065002 (2009)
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)
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)
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)
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
Wang, T.-Y., Cai, X.Q., Ren, Y.L., Zhang, R.L.: Security of quantum digital signature. Sci. Rep. 5, 9231 (2015)
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)
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)
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)
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)
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)
Gao, F., Liu, B., Wen, Q.-Y.: Flexible quantum private queries based on quantum key distribution. Opt. Exp. 20, 17411–17420 (2012)
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)
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)
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)
Diffie, W., Hellman, M.: New directions in cryptography. IEEE Trans. Inf. Theory 22, 644–654 (1976)
Zhou, N., Zeng, G., Xiong, J.: Quantum key agreement protocol. Electron. Lett. 40(18), 1149 (2004)
Tsai, C.W., Hwang, T.: On quantum key agreement protocol, Technical Report, CS-I-E, NCKU, Taiwan, ROC, 2009
Chong, S.K., Hwang, T.: Quantum key agreement protocol based on BB84. Opt. Commun. 283, 1192–1195 (2010)
He, Y.F., Ma, W.P.: Two-party quantum key agreement with five-particle entangled states. Int. J. Quantum Inf. 15(03), 1750018 (2017)
Cai, B., Guo, G., Lin, S.: Multi-party quantum key agreement with teleportation. Mod. Phys. Lett. B 31(10), 1750102 (2017)
Cao, H., Ma, W.: Multiparty quantum key agreement based on quantum search algorithm. Sci. Rep. 7, 45046 (2017)
Shi, R.H., Zhong, H.: Multi-party quantum key agreement with Bell states and Bell measurements. Quantum Inf. Process. 12, 921–932 (2013)
Liu, B., Gao, F., Huang, W., Wen, Q.-Y.: Multiparty quantum key agreement with single particles. Quantum Inf. Process. 12, 1797–1805 (2013)
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)
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)
Sun, Z.W., Yu, J.P., Wang, P.: Efficient multi-party quantum key agreement by cluster States. Quantum Inf. Process. 15, 373–384 (2016)
He, Y.-F., Ma, W.-P.: Quantum key agreement protocols with four-qubit cluster states. Quantum Inf. Process. 14, 3483–3498 (2015)
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)
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)
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)
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)
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)
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)
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)
Huang, W., Su, Q., Wu, X., Li, Y.B., Sun, Y.: Quantum key agreement against collective decoherence. Int. J. Theor. Phys. 53, 2891 (2014)
He, Y.F., Ma, W.P.: Two-party quantum key agreement against collective noise. Quantum Inf. Process. 15, 5023–5035 (2016)
He, Y.-F., Ma, W.-P.: Two quantum key agreement protocols immune to collective noise. Int. J. Theor. Phys. 56, 328–338 (2017)
Gao, H., Chen, X.G., Qian, S.R.: Two-party quantum key agreement protocols under collective noise channel. Quantum Inf. Process. 17, 140 (2018)
Gisin, N., Fasel, S., Kraus, B., Zbinden, H., Ribordy, G.: Trojan-horse attacks on quantum-key distribution systems. Phys. Rev. A 73, 022320 (2006)
Cai, Q.Y.: Eavesdropping on the two-way quantum communication protocols with invisible photons. Phys. Lett. A 351, 23–25 (2006)
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)
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)
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
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)
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)
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)
Cabello, A.: Quantum key distribution in the Holevo limit. Phys. Rev. Lett. 85, 5635–5638 (2000)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Ma, W.-X.: Conservation laws by symmetries and adjoint symmetries. Discrete. Contin. Dyn. Syst. Ser. S 11(4), 707–721 (2018)
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)
Ma, W.-X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)
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)
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)
Liu, F.: Continuity and approximate differentiability of multisublinear fractional maximal functions. Math. Inequal. Appl. 21(1), 25–40 (2018)
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)
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)
Xu, X.-X., Sun, Y.-P.: Two symmetry constraints for a generalized Dirac integrable hierarchy. J. Math. Anal. Appl. 458, 1073–1090 (2018)
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)
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)
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)
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)
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)
Liu, F.: A remark on the regularity of the discrete maximal operator. Bull. Aust. Math. Soc. 95, 108–120 (2017)
Liu, F.: Integral operators of Marcinkiewicz type on Triebel-Lizorkin spaces. Math. Nachr. 290, 75–96 (2017)
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)
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)
Liu, F., Wu, H.X.: On the regularity of maximal operators supported by submanifolds. J. Math. Anal. Appl. 453, 144–158 (2017)
Liu, F., Wu, H.X.: Regularity of discrete multisublinear fractional maximal functions. Sci. China Math. 60(8), 1461–1476 (2017)
Liu, F., Wu, H.X.: Endpoint regularity of multisublinear fractional maximal functions. Can. Math. Bull. 60(3), 586–603 (2017)
Liu, F., Mao, S.Z.: On the regularity of the one-sided Hardy-Littlewood maximal functions. Czech. Math. J. 67(142), 219–234 (2017)
Liu, F.: On the Triebel-Lizorkin space boundedness of Marcinkiewicz integrals along compound surfaces. Math. Inequal. Appl. 20(2), 515–535 (2017)
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)
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)
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)
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)
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)
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)
Zhang, J.B., Ma, W.X.: Mixed lump-kink solutions to the BKP equation. Comput. Math. Appl. 74, 591–596 (2017)
Zhao, H.Q., Ma, W.X.: Mixed lump-kink solutions to the KP equation. Comput. Math. Appl. 74, 1399–1405 (2017)
Liu, F., Wu, H.X.: Singular integrals related to homogeneous mappings in Triebel-Lizorkin spaces. J. Math. Inequal. 11(4), 1075–1097 (2017)
Liu, F.: Rough singular integrals associated to surfaces of revolution on Triebel-Lizorkin spaces. Rocky Mt. J. Math. 47(5), 1617–1653 (2017)
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)
Wang, Y.H.: Beyond regular semigroups. Semigroup Forum 92(2), 414–448 (2016)
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)
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)
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)
Cui, Y.J.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)
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)
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)
Liu, F., Mao, S.Z., Wu, H.X.: On rough singular integrals related to homogeneous mappings. Collectanea Math. 67(1), 113–132 (2016)
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)
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)
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)
Cui, Y.J.: Existence of solutions for coupled integral boundary value problem at resonance. Publ. Mathe-Debrecen 89(1–2), 73–88 (2016)
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)
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)
Liu, F., Wu, H.X.: L-p bounds for marcinkiewicz integrals associated to homogeneous mappings. Monatshefte für Mathematik 181(4), 875–906 (2016)
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)
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)
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)
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)
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)
Liu, F., Zhang, D.Q.: Parabolic marcinkiewicz integrals associated to polynomials compound curves and extrapolation. Bull. Korean Math. Soc. 52(3), 771–788 (2015)
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)
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)
Liu, F., Wu, H.X.: On the regularity of the multisublinear maximal functions. Can. Math. Bull. 58(4), 808–817 (2015)
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)
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)
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)
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)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-019-2187-8