Abstract
In this article, we proposed a new chaotic map and is compared with existing chaotic maps such as Logistic map and Tent map. The value of maximal Lyapunov exponent of the proposed chaotic map goes beyond 1 and shows more chaotic behaviour than existing one-dimensional chaotic maps. This shows that proposed chaotic maps are more effective for cryptographic applications. Further, we are using one-dimensional chaotic maps to generate random time series data and define a method to create a network. Lyapunov exponent and entropy of the data are considered to measure the randomness or chaotic behaviour of the time series data. We study the relationship between concurrence (for the two-qubit quantum states) and Lyapunov exponent with respect to initial condition and parameter of the logistic map which is showing how chaos can lead to concurrence based on such Lyapunov exponents.
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References
M. Berezowski, M. Lawnik, Identification of fast-changing signals by means of adaptive chaotic transformations (2016), arXiv:1603.06763
M. Lawnik, M. Berezowski, Identification of the oscillation period of chemical reactors by chaotic sampling of the conversion degree. Chemical and Process Engineering 35(3), 387–393 (2014)
S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, 2018)
W.F.H. Al-Shameri, M.A. Mahiub, Some dynamical properties of the family of tent maps. Int. J. Math. Anal. 7(29), 1433–1449 (2013)
L. Shan, H. Qiang, J. Li, Z. Wang, Chaotic optimization algorithm based on tent map. Control Decis. 20(2), 179–182 (2005)
T. Yoshida, H. Mori, H. Shigematsu, Analytic study of chaos of the tent map: band structures, power spectra, and critical behaviors. J. Stat. Phys. 31(2), 279–308 (1983)
L. Kocarev, G. Jakimoski, Logistic map as a block encryption algorithm. Phys. Lett. A 289(4–5), 199–206 (2001)
S.C. Phatak, S. Suresh Rao, Logistic map: a possible random-number generator. Phys. Rev. E 51(4), 3670 (1995)
M.S. Baptista, Cryptography with chaos. Phys. Lett. A 240(1–2), 50–54 (1998)
N.K. Pareek, V. Patidar, K.K. Sud, Image encryption using chaotic logistic map. Image Vis. Comput. 24(9), 926–934 (2006)
T.P. Spiller, Quantum information processing: cryptography, computation, and teleportation. Proc. IEEE 84(12), 1719–1746 (1996)
C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without bell’s theorem. Phys. Rev. Lett. 68(5), 557 (1992)
P.W. Shor, J. Preskill, Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85(2), 441 (2000)
A.K. Ekert, Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67(6), 661 (1991)
P.C. Müller, Calculation of lyapunov exponents for dynamic systems with discontinuities. Chaos, Solitons, Fractals, 5(9), 1671–1681 (1995)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining lyapunov exponents from a time series. Phys. D: Nonlinear Phenom. 16(3), 285–317 (1985)
C. Jin, H. Liu, A color image encryption scheme based on arnold scrambling and quantum chaotic. IJ Netw. Secur. 19(3), 347–357 (2017)
A.G. Radwan, S.K. Abd-El-Hafiz, Image encryption using generalized tent map, in 2013 IEEE 20th International Conference on Electronics, Circuits, and Systems (ICECS) (IEEE, 2013), pp. 653–656
X. Zhang, Y. Cao, A novel chaotic map and an improved chaos-based image encryption scheme. Sci. World J. 2014, (2014)
E. Ceyhan, Edge density of new graph types based on a random digraph family. Stat. Methodol. 33, 31–54 (2016)
Pradumn Kumar Pandey and Bibhas Adhikari, Context dependent preferential attachment model for complex networks. Phys. A: Stat. Mech. Its Appl. 436, 499–508 (2015)
R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81(2), 865 (2009)
W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)
Acknowledgements
The authors are grateful to Satish Sangwan for valuable comments and suggestions.
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Joshi, A., Kumar, A. (2020). Chaotic Maps: Applications to Cryptography and Network Generation for the Graph Laplacian Quantum States. In: Deo, N., Gupta, V., Acu, A., Agrawal, P. (eds) Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory. ICRAPAM 2018. Springer Proceedings in Mathematics & Statistics, vol 307. Springer, Singapore. https://doi.org/10.1007/978-981-15-1157-8_14
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DOI: https://doi.org/10.1007/978-981-15-1157-8_14
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