A Novel Triangular Chaotic Map (TCM) with Full Intensive Chaotic Population Based on Logistic Map
Author(s)
Mahmoud Maqableh
Affiliation(s)
Management Information Systems, Faculty of Business, The University of Jordan, Amman, Jordan.
Management Information Systems, Faculty of Business, The University of Jordan, Amman, Jordan.
ABSTRACT
Chaos theory attempts to explain the result of a system that is sensitive to initial conditions, complex, and shows an unpredictable behaviour. Chaotic systems are sensitive to any change or changes in the initial condition(s) and are unpredictable in the long term. Chaos theory are implementing today in many different fields of studies. In this research, we propose a new one-dimensional Triangular Chaotic Map (TCM) with full intensive chaotic population. TCM chaotic map is a one-way function that prevents the finding of a relationship between the successive output values and increases the randomness of output results. The tests and analysis results of the proposed triangular chaotic map show a great sensitivity to initial conditions, have unpredictability, are uniformly distributed and random-like and have an infinite range of intensive chaotic population with large positive Lyapunov exponent values. Moreover, TCM characteristics are very promising for possible utilization in many different study fields.
Chaos theory attempts to explain the result of a system that is sensitive to initial conditions, complex, and shows an unpredictable behaviour. Chaotic systems are sensitive to any change or changes in the initial condition(s) and are unpredictable in the long term. Chaos theory are implementing today in many different fields of studies. In this research, we propose a new one-dimensional Triangular Chaotic Map (TCM) with full intensive chaotic population. TCM chaotic map is a one-way function that prevents the finding of a relationship between the successive output values and increases the randomness of output results. The tests and analysis results of the proposed triangular chaotic map show a great sensitivity to initial conditions, have unpredictability, are uniformly distributed and random-like and have an infinite range of intensive chaotic population with large positive Lyapunov exponent values. Moreover, TCM characteristics are very promising for possible utilization in many different study fields.
Cite this paper
Maqableh, M. (2015) A Novel Triangular Chaotic Map (TCM) with Full Intensive Chaotic Population Based on Logistic Map. Journal of Software Engineering and Applications, 8, 635-659. doi: 10.4236/jsea.2015.812059.
Maqableh, M. (2015) A Novel Triangular Chaotic Map (TCM) with Full Intensive Chaotic Population Based on Logistic Map. Journal of Software Engineering and Applications, 8, 635-659. doi: 10.4236/jsea.2015.812059.
References
[1] Sneyers, R. (1997) Climate Chaotic Instability: Statistical Determination and Theoretical Background. Environmetrics, 8, 517-532.
http://dx.doi.org/10.1002/(SICI)1099-095X(199709/10)8:5<517::AID-ENV267>3.0.CO;2-L [2] Zeng, X., Pielke, R.A. and Eykholt, R. (1993) Chaos Theory and Its Application to the Atmosphere. Bulletin of the American Meteorological Society, 74, 631-639.
http://dx.doi.org/10.1175/1520-0477(1993)074<0631:CTAIAT>2.0.CO;2 [3] Wikipedia. Chaos Theory. Cited 17 January 2009.
http://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=264934743 [4] Serletis, A. and Gogas, P. (2000) Purchasing Power Parity, Nonlinearity and Chaos. Applied Financial Economics, 10, 615-622.
http://dx.doi.org/10.1080/096031000437962 [5] Serletis, A. and Gogas, P. (1997) Chaos in East European Black Market Exchange Rates. Research in Economics, 51, 359-385.
http://dx.doi.org/10.1006/reec.1997.0050 [6] Wikipedia (2015) Chaos Theory.
https://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=693847517 [7] Maqableh, M., Samsudin, A.B. and Alia, M.A. (2008) New Hash Function Based on Chaos Theory (CHA-1). IJCSNS International Journal of Computer Science and Network Security, 8, 20-26. [8] Kocarev, L. (2001) Chaos-Based Cryptography: A Brief Overview. IEEE Circuits and Systems Magazine, 1, 6-21.
http://dx.doi.org/10.1109/7384.963463 [9] Pareek, N.K., Patidar, V. and Sud, K.K. (2003) Discrete Chaotic Cryptography Using External Key. Physics Letters A, 309, 75-82.
http://dx.doi.org/10.1016/S0375-9601(03)00122-1 [10] Pareek, N.K., Patidar, V. and Sud, K.K. (2005) Cryptography Using Multiple One-Dimensional Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 10, 715-723.
http://dx.doi.org/10.1016/j.cnsns.2004.03.006 [11] Xiang, T., Liao, X.F., Tang, G.P., Chen, Y. and Wong, K.W. (2006) A Novel Block Cryptosystem Based on Iterating a Chaotic Map. Physics Letters A, 349, 109-115.
http://dx.doi.org/10.1016/j.physleta.2005.02.083 [12] Chen, S., Zhong, X.X. and Wu, Z.Z. (2008) Chaos Block Cipher for Wireless Sensor Network. Science in China Series F: Information Sciences, 51, 1055-1063.
http://dx.doi.org/10.1007/s11432-008-0102-5 [13] Peng, J., You, M.Y., Yang, Z.M. and Jin, S.Z. (2007) Research on a Block Encryption Cipher Based on Chaotic Dynamical System. 3rd International Conference on Natural Computation, ICNC 2007, Haikou, 24-27 August 2007, 744-748.
http://dx.doi.org/10.1109/ICNC.2007.612 [14] Yang, H.Q., Liao, X.F., Wong, K.W., Zhang, W. and Wei, P.C. (2007) A New Block Cipher Based on Chaotic Map and Group Theory. Chaos, Solitons & Fractals, 40, 50-59.
http://dx.doi.org/10.1016/j.chaos.2007.07.056 [15] Peng, J., Jin, S.Z., Chen, G.R., Yang, Z.M. and Liao, X.F. (2008) An Image Encryption Scheme Based on Chaotic Map. 4th International Conference on Natural Computation, ICNC ’08, Jinan, 18-20 October 2008, 595-599.
http://dx.doi.org/10.1109/icnc.2008.227 [16] Lian, S. (2009) A Block Cipher Based on Chaotic Neural Networks. Neurocomputing, 72, 1296-1301.
http://dx.doi.org/10.1016/j.neucom.2008.11.005 [17] Wang, F.J., Zhang, Y.P. and Cao, T.J. (2009) Research of Chaotic Block Cipher Algorithm Based on Logistic Map. 2nd International Conference on Intelligent Computation Technology and Automation, 2009, ICICTA ’09, Changsha, 10-11 October 2009, 678-681.
http://dx.doi.org/10.1109/ICICTA.2009.169 [18] Peng, J., Jin, S.Z., Liu, H.L. and Liu, Y.G. (2009) A Block Cipher Based on a Hybrid of Chaotic System and Feistel Network. 5th International Conference on Natural Computation, ICNC ’09, Tianjin, 14-16 August 2009, 427-431.
http://dx.doi.org/10.1109/icnc.2009.663 [19] Amin, M., Faragallah, O.S. and Abd El-Latif, A.A. (2010) A Chaotic Block Cipher Algorithm for Image Cryptosystems. Communications in Nonlinear Science and Numerical Simulation, 15, 3484-3497.
http://dx.doi.org/10.1016/j.cnsns.2009.12.025 [20] Huang, J.-H. and Liu, Y. (2010) A Block Encryption Algorithm Combined with the Logistic Mapping and SPN Structure. 2010 2nd International Conference on Industrial and Information Systems (IIS), Dalian, 10-11 July 2010, 156-159.
http://dx.doi.org/10.1109/indusis.2010.5565655 [21] Zhao, G., Chen, G.R., Fang, J.Q. and Xu, G. (2011) Block Cipher Design: Generalized Single-Use-Algorithm Based on Chaos. Tsinghua Science & Technology, 16, 194-206.
http://dx.doi.org/10.1016/S1007-0214(11)70030-X [22] Masuda, N., Jakimoski, G., Aihara, K. and Kocarev, L. (2006) Chaotic Block Ciphers: From Theory to Practical Algorithms. IEEE Transactions on Circuits and Systems I: Regular Papers, 53, 1341-1352.
http://dx.doi.org/10.1109/TCSI.2006.874182 [23] Habutsu, T., Nishio, Y., Sasase, I. and Mori, S. (1991) A Secret Key Cryptosystem by Iterating a Chaotic Map. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 127-140.
http://dx.doi.org/10.1007/3-540-46416-6_11 [24] Gutowitz, H.A. (1993) Cryptography with Dynamical Systems, in Cellular Automata and Cooperative Phenomena. Kluwer Academic Press, Dordrecht.
http://dx.doi.org/10.1007/978-94-011-1691-6_21 [25] Kotulski, Z. and Szczepanski, J. (1997) Discrete Chaotic Cryptography (DCC). Annalen der Physik, 6, 381-394.
http://dx.doi.org/10.1002/andp.19975090504 [26] García, P. and Jiménez, J. (2002) Communication through Chaotic Map Systems. Physics Letters A, 298, 35-40.
http://dx.doi.org/10.1016/S0375-9601(02)00382-1 [27] Masuda, N. and Aihara, K. (2002) Cryptosystems with Discretized Chaotic Maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49, 28-40.
http://dx.doi.org/10.1109/81.974872 [28] Baptista, M.S. (1998) Cryptography with Chaos. Physics Letters A, 240, 50-54.
http://dx.doi.org/10.1016/S0375-9601(98)00086-3 [29] Wong, W.-K., Lee, L.-P. and Wong, K.-W. (2001) A Modified Chaotic Cryptographic Method. Computer Physics Communications, 138, 234-236.
http://dx.doi.org/10.1016/S0010-4655(01)00220-X [30] Alvarez, E., Fernández, A., Garcia, P., Jiménez, J. and Marcano, A. (1999) New Approach to Chaotic Encryption. Physics Letters A, 263, 373-375.
http://dx.doi.org/10.1016/S0375-9601(99)00747-1 [31] Wong, K.W. (2002) A Fast Chaotic Cryptographic Scheme with Dynamic Look-Up Table. Physics Letters A, 298, 238-242.
http://dx.doi.org/10.1016/S0375-9601(02)00431-0 [32] Machado, R.F., Baptista, M.S. and Grebogi, C. (2004) Cryptography with Chaos at the Physical Level. Chaos, Solitons & Fractals, 21, 1265-1269.
http://dx.doi.org/10.1016/j.chaos.2003.12.094 [33] Guan, Z.-H., Huang, F. and Guan, W. (2005) Chaos-Based Image Encryption Algorithm. Physics Letters A, 346, 153-157.
http://dx.doi.org/10.1016/j.physleta.2005.08.006 [34] Gao, T. and Chen, Z. (2008) Image Encryption Based on a New Total Shuffling Algorithm. Chaos, Solitons & Fractals, 38, 213-220.
http://dx.doi.org/10.1016/j.chaos.2006.11.009 [35] Wong, K.W. (2003) A Combined Chaotic Cryptographic and Hashing Scheme. Physics Letters A, 307, 292-298.
http://dx.doi.org/10.1016/S0375-9601(02)01770-X [36] Xiao, D., Liao, X. and Deng, S. (2005) One-Way Hash Function Construction Based on the Chaotic Map with Changeable-Parameter. Chaos, Solitons & Fractals, 24, 65-71. http://dx.doi.org/10.1016/S0960-0779(04)00456-4 [37] Lian, S.G., Liu, Z.X., Ren, Z. and Wang, H.L. (2006) Hash Function Based on Chaotic Neural Networks. 2006 IEEE International Symposium on Circuits and Systems, ISCAS 2006, Island of Kos, 21-24 May 2006.
http://dx.doi.org/10.1109/iscas.2006.1692566 [38] Peng, F. and Qiu, S.-S. (2007) One-Way Hash Functions Based on Iterated Chaotic Systems. International Conference on Communications, Circuits and Systems, ICCCAS 2007, Kokura, 11-13 July 2007, 1070-1074.
http://dx.doi.org/10.1109/ICCCAS.2007.4348231 [39] Khan, M.K., Zhang, J. and Wang, X. (2008) Chaotic Hash-Based Fingerprint Biometric Remote User Authentication Scheme on Mobile Devices. Chaos, Solitons & Fractals, 35, 519-524.
http://dx.doi.org/10.1016/j.chaos.2006.05.061 [40] Xiao, D., Liao, X. and Deng, S. (2008) Parallel Keyed Hash Function Construction Based on Chaotic Maps. Physics Letters A, 372, 4682-4688.
http://dx.doi.org/10.1016/j.physleta.2008.04.060 [41] Song, Y.R. and Jiang, G.P. (2008) Hash Function Construction Based on Chaotic Coupled Map Network. The 9th International Conference for Young Computer Scientists, ICYCS 2008, Hunan, 18-21 November 2008, 2753-2758.
http://dx.doi.org/10.1109/ICYCS.2008.134 [42] Deng, S., Xiao, D., Li, Y.T. and Peng, W.B. (2009) A Novel Combined Cryptographic and Hash Algorithm Based on Chaotic Control Character. Communications in Nonlinear Science and Numerical Simulation, 14, 3889-3900.
http://dx.doi.org/10.1016/j.cnsns.2009.02.020 [43] Guyeux, C. and Bahi, J.M. (2010) Topological Chaos and Chaotic Iterations Application to Hash Functions. The 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, 18-23 July 2010, 1-7.
http://dx.doi.org/10.1109/ijcnn.2010.5596512 [44] Huang, Z. (2011) A More Secure Parallel Keyed Hash Function Based on Chaotic Neural Network. Communications in Nonlinear Science and Numerical Simulation, 16, 3245-3256. http://dx.doi.org/10.1016/j.cnsns.2010.12.009 [45] Wang, Y., Wong, K.-W. and Xiao, D. (2011) Parallel Hash Function Construction Based on Coupled Map Lattices. Communications in Nonlinear Science and Numerical Simulation, 16, 2810-2821.
http://dx.doi.org/10.1016/j.cnsns.2010.10.001 [46] Yi, X. (2005) Hash Function Based on Chaotic Tent Maps. IEEE Transactions on Circuits and Systems II: Express Briefs, 52, 354-357.
http://dx.doi.org/10.1109/TCSII.2005.848992 [47] Zhang, J., Wang, X. and Zhang, W. (2007) Chaotic Keyed Hash Function Based on Feedforward-Feedback Nonlinear Digital Filter. Physics Letters A, 362, 439-448.
http://dx.doi.org/10.1016/j.physleta.2006.10.052 [48] Amin, M., Faragallah, O.S. and Abd El-Latif, A.A. (2009) Chaos-Based Hash Function (CBHF) for Cryptographic Applications. Chaos, Solitons & Fractals, 42, 767-772.
http://dx.doi.org/10.1016/j.chaos.2009.02.001 [49] Xiao, D., Shih, F.Y. and Liao, X. (2010) A Chaos-Based Hash Function with both Modification Detection and Localization Capabilities. Communications in Nonlinear Science and Numerical Simulation, 15, 2254-2261.
http://dx.doi.org/10.1016/j.cnsns.2009.10.012 [50] Kocarev, L. and Jakimoski, G. (2003) Pseudorandom Bits Generated by Chaotic Maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 123-126.
http://dx.doi.org/10.1109/TCSI.2002.804550 [51] Tong, X.-J., Cui, M.-G. and Jiang, W. (2006) The Production Algorithm of Pseudo-Random Number Generator Based on Compound Non-Linear Chaos System. International Conference on Intelligent Information Hiding and Multimedia Signal Processing, 2006, IIH-MSP ’06, Pasadena, 18-20 December 2006, 685-688.
http://dx.doi.org/10.1109/IIH-MSP.2006.265094 [52] Chen, S. and Zhong, X.-X. (2007) Chaotic Block Iterating Method for Pseudo-Random Sequence Generator. The Journal of China Universities of Posts and Telecommunications, 14, 45-48.
http://dx.doi.org/10.1016/s1005-8885(07)60054-5 [53] Zheng, F., Tian, X.-J., Song, J.-Y. and Li, X.-Y. (2008) Pseudo-Random Sequence Generator Based on the Generalized Henon Map. The Journal of China Universities of Posts and Telecommunications, 15, 64-68.
http://dx.doi.org/10.1016/S1005-8885(08)60109-0 [54] Qi, A.X., Han, C.Y. and Wang, G.Y. (2010) Design and FPGA Realization of a Pseudo Random Sequence Generator Based on a Switched Chaos. International Conference on Communications, Circuits and Systems (ICCCAS), Chengdu, 28-30 July 2010, 417-420.
http://dx.doi.org/10.1109/ICCCAS.2010.5581965 [55] Yoon, J.W. and Kim, H. (2010) An Image Encryption Scheme with a Pseudorandom Permutation Based on Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 3998-4006.
http://dx.doi.org/10.1016/j.cnsns.2010.01.041 [56] Dabal, P. and Pelka, R. (2011) A Chaos-Based Pseudo-Random Bit Generator Implemented in FPGA Device. 2011 IEEE 14th International Symposium on Design and Diagnostics of Electronic Circuits & Systems (DDECS), Cottbus, 13-15 April 2011, 151-154.
http://dx.doi.org/10.1109/ddecs.2011.5783069 [57] Forré, R. (1991) The Hénon Attractor as a Keystream Generator. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 76-81. [58] Matthews, R.A.J. (1989) On the Derivation of a “Chaotic” Encryption Algorithm. Cryptologia, 13, 29-42.
http://dx.doi.org/10.1080/0161-118991863745 [59] Li, S., Mou, X. and Cai, Y. (2001) Improving Security of a Chaotic Encryption Approach. Physics Letters A, 290, 127-133.
http://dx.doi.org/10.1016/S0375-9601(01)00612-0 [60] Wolfram, S. (1985) Cryptography with Cellular Automata. In: Williams, H.C., Ed., Advances in Cryptology— CRYPTO ’85 Proceedings, Lecture Notes in Computer Science, Spinger-Verlag, Berlin, 429-432. [61] Lee, P.-H., Pei, S.-C. and Chen, Y.-Y. (2003) Generating Chaotic Stream Ciphers Using Chaotic Systems. Chinese Journal of Physics, 41, 559-581. [62] Sang, T., Wang, R. and Yan, Y. (2000) Constructing Chaotic Discrete Sequences for Digital Communications Based on Correlation Analysis. IEEE Transactions on Signal Processing, 48, 2557-2565.
http://dx.doi.org/10.1109/78.863058 [63] Kwok, H.S. and Tang, W.K.S. (2007) A Fast Image Encryption System Based on Chaotic Maps with Finite Precision Representation. Chaos, Solitons & Fractals, 32, 1518-1529.
http://dx.doi.org/10.1016/j.chaos.2005.11.090 [64] Patidar, V., Pareek, N.K. and Sud, K.K. (2009) A New Substitution-Diffusion Based Image Cipher Using Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 14, 3056-3075.
http://dx.doi.org/10.1016/j.cnsns.2008.11.005 [65] Patidar, V., Pareek, N.K., Purohit, G. and Sud, K.K. (2010) Modified Substitution-Diffusion Image Cipher Using Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 2755-2765.
http://dx.doi.org/10.1016/j.cnsns.2009.11.010 [66] Biham, E. (1991) Cryptanalysis of the Chaotic-Map Cryptosystem Suggested at EUROCRYPT’91. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 532-534.
http://dx.doi.org/10.1007/3-540-46416-6_49 [67] Wheeler, D.D. (1989) Problems with Chaotic Cryptosystems. Cryptologia, 13, 243-250.
http://dx.doi.org/10.1080/0161-118991863934 [68] Alvarez, G., Montoya, F., Romera, M. and Pastor, G. (2000) Cryptanalysis of a Chaotic Encryption System. Physics Letters A, 276, 191-196.
http://dx.doi.org/10.1016/S0375-9601(00)00642-3 [69] Alvarez, G., Montoya, F., Romera, M. and Pastor, G. (2003) Cryptanalysis of a Discrete Chaotic Cryptosystem Using External Key. Physics Letters A, 319, 334-339.
http://dx.doi.org/10.1016/j.physleta.2003.10.044 [70] Wei, J., Liao, X.F., Wong, K.W. and Zhou, T. (2007) Cryptanalysis of a Cryptosystem Using Multiple One-Dimensional Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 12, 814-822.
http://dx.doi.org/10.1016/j.cnsns.2005.06.001 [71] Li, C.Q., Li, S.J., Alvarez, G., Chen, G.R. and Lo, K.-T. (2008) Cryptanalysis of a Chaotic Block Cipher with External Key and Its Improved Version. Chaos, Solitons & Fractals, 37, 299-307.
http://dx.doi.org/10.1016/j.chaos.2006.08.025 [72] Wang, X. and Yu, C. (2009) Cryptanalysis and Improvement on a Cryptosystem Based on a Chaotic Map. Computers & Mathematics with Applications, 57, 476-482.
http://dx.doi.org/10.1016/j.camwa.2008.09.042 [73] Yang, J., Xiao, D. and Xiang, T. (2011) Cryptanalysis of a Chaos Block Cipher for Wireless Sensor Network. Communications in Nonlinear Science and Numerical Simulation, 16, 844-850.
http://dx.doi.org/10.1016/j.cnsns.2010.05.005 [74] álvarez, G., Montoya, F., Romera, M. and Pastor, G. (2004) Keystream Cryptanalysis of a Chaotic Cryptographic Method. Computer Physics Communications, 156, 205-207.
http://dx.doi.org/10.1016/S0010-4655(03)00432-6 [75] Jakimoski, G. and Kocarev, L. (2001) Analysis of Some Recently Proposed Chaos-Based Encryption Algorithms. Physics Letters A, 291, 381-384.
http://dx.doi.org/10.1016/S0375-9601(01)00771-X [76] Li, S.J., Mou, X.Q., Ji, Z., Zhang, J.H. and Cai, Y.L. (2003) Performance Analysis of Jakimoski-Kocarev Attack on a Class of Chaotic Cryptosystems. Physics Letters A, 307, 22-28.
http://dx.doi.org/10.1016/S0375-9601(02)01659-6 [77] ?okal, C. and Solak, E. (2009) Cryptanalysis of a Chaos-Based Image Encryption Algorithm. Physics Letters A, 373, 1357-1360.
http://dx.doi.org/10.1016/j.physleta.2009.02.030 [78] Maqableh, M.M. and Dantchev, S. (2009) Cryptanalysis of Chaos-Based Hash Function (CBHF). 1st International Alternative Workshop on Aggressive Computing and Security—iAWACS, Laval, 23-25 October 2009, 20-30. [79] Yang, Q.-T., Gao, T.-G., Fan, L. and Gu, Q.-L. (2009) Analysis of One-Way Alterable Length Hash Function Based on Cell Neural Network. 5th International Conference on Information Assurance and Security, IAS ’09, Xi’an, 18-20 August 2009, 391-395.
http://dx.doi.org/10.1109/ias.2009.87 [80] Xiao, D., Liao, X. and Wang, Y. (2009) Improving the Security of a Parallel Keyed Hash Function Based on Chaotic Maps. Physics Letters A, 373, 4346-4353.
http://dx.doi.org/10.1016/j.physleta.2009.09.059 [81] Deng, S., Li, Y. and Xiao, D. (2010) Analysis and Improvement of a Chaos-Based Hash Function Construction. Communications in Nonlinear Science and Numerical Simulation, 15, 1338-1347.
http://dx.doi.org/10.1016/j.cnsns.2009.05.065 [82] Li, C.Q., Li, S.J., Chen, G.R. and Halang, W.A. (2009) Cryptanalysis of an Image Encryption Scheme Based on a Compound Chaotic Sequence. Image and Vision Computing, 27, 1035-1039.
http://dx.doi.org/10.1016/j.imavis.2008.09.004 [83] Rhouma, R., Solak, E. and Belghith, S. (2010) Cryptanalysis of a New Substitution-Diffusion Based Image Cipher. Communications in Nonlinear Science and Numerical Simulation, 15, 1887-1892.
http://dx.doi.org/10.1016/j.cnsns.2009.07.007 [84] Li, C., Li, S. and Lo, K.-T. (2011) Breaking a Modified Substitution-Diffusion Image Cipher Based on Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 16, 837-843.
http://dx.doi.org/10.1016/j.cnsns.2010.05.008 [85] Maqableh, M.M. (2011) Fast Parallel Keyed Hash Functions Based on Chaotic Maps (PKHC). Western European Workshop on Research in Cryptology, Weimar, 20-22 July 2011, 33-40. [86] Maqableh, M.M. (2010) Secure Hash Functions Based on Chaotic Maps for E-Commerce Application. International Journal of Information Technology and Management information System (IJITMIS), 1, 12-19. [87] Maqableh, M.M. (2010) Fast Hash Function Based on BCCM Encryption Algorithm for E-Commerce (HFBCCM). 5th International Conference on e-Commerce in Developing Countries: With Focus on Export, Le Havre, 15-16 September 2010, 55-64. [88] Bertuglia, C.S. and Vaio, F. (2005) Nonlinearity, Chaos & Complexity: The Dynamics of Natural and Social Systems. Oxford University Press. [89] Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996) Chaos an Introduction to Dynamical Systems. Springer-Verlag, New York. [90] Solari, H.G., Natiello, M.A. and Mindlin, G.B. (1996) Nonlinear Dynamics A Two-Way Trip from Physics to Math. Institute of Physics Publishing, Bristol. [91] Baker, G.L. and Gollub, J.P. (1990) Chaotic Dynamics an Introduction. Press Syndicate of the University of Cambridge, New York. [92] Poincaré, J.H. (1890) Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt. Acta Mathematica, 13, 1-270. [93] Lorenz, E.N. (1963) Deterministic Nonperiodic Flow. Journal of Atmospheric Sciences, 20, 130-141.
http://dx.doi.org/10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2 [94] Pritchard, J. (1996) The Chaos Cookbook. Butterworh-Heinemann, Oxford. [95] Rahimi, A., Mohammadi, S. and Rahimi, R. (2009) An Efficient Iris Authentication Using Chaos Theory-Based Cryptography for E-Commerce Transactions. International Conference for Internet Technology and Secured Transactions (ICITST 2009). [96] Serletis, A. and Gogas, P. (1999) The North American Natural Gas Liquids Markets Are Chaotic. The Energy Journal, 20, 83-103.
http://dx.doi.org/10.5547/ISSN0195-6574-EJ-Vol20-No1-5 [97] Gilmore, R. (2004) Chaos and Attractors. Encyclopedia of Mathematical Physics, EMP MS 93. [98] Tullaro, N.B., Abbott, T. and Reilly, J.P. (1992) An Experimental Approach to Nonlinear Dynamics and Chaos. Vol. 1, Addison-Wesley, Boston. [99] Parker, T.S. and Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-3486-9 [100] Schmitz, R. (2001) Use of Chaotic Dynamical Systems in Cryptography. Journal of the Franklin Institute, 338, 429-441.
http://dx.doi.org/10.1016/S0016-0032(00)00087-9 [101] Maqableh, M.M. (2012) Analysis and Design Security Primitives Based on Chaotic Systems for eCommerce. Doctoral Thesis, Durham University, Durham. [102] Grassi, G. and Mascoio, S. (1999) Synchronizing Hyperchaotic Systems by Observer Design. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 46, 478-483.
http://dx.doi.org/10.1109/82.755422 [103] May, R.M. (1976) Simple Mathematical Models with Very Complicated Dynamics. Nature, 261, 459-467.
http://dx.doi.org/10.1038/261459a0 [104] Wikipedia. Logistic Map. Cited 20 February 2009.
http://en.wikipedia.org/w/index.php?title=chaotic_maps&oldid=261864353 [105] Naess, A. (2000) Chaos and Nonlinear Stochastic Dynamics. Probabilistic Engineering Mechanics, 15, 37-47.
http://dx.doi.org/10.1016/S0266-8920(99)00007-7 [106] Contributors, W. (2005) Chaos Theory.
http://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=264934743 [107] Rossler, O. (1976) An Equation for Continous Chaos. Physics Letters A, 57, 397-398.
http://dx.doi.org/10.1016/0375-9601(76)90101-8 [108] Ho, A. (2006) Chaos Introduction.
http://www.zeuscat.com/andrew/chaos/chaos.html [109] Wikipedia. Tent Map. Cited 25 February 2009.
http://en.wikipedia.org/w/index.php?title=Tent_map&oldid=186656075 [110] Li, S., Chen, G. and Mou, X. (2005) On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps. International Journal of Bifurcation and Chaos, 15, 3119-3151. http://dx.doi.org/10.1142/S0218127405014052 [111] Jun, P., Jin, S.Z., Liu, Y.G., Yang, Z.M., You, M.Y. and Pei, Y.J. (2008) A Novel Scheme for Image Encryption Based on Piecewise Linear Chaotic Map. 2008 IEEE Conference on Cybernetics and Intelligent Systems, Chengdu, 21-24 September 2008, 1012-1016. [112] Rhouma, R., Arroyo, D. and Belghith, S. (2009) A New Color Image Cryptosystem Based on a Piecewise Linear Chaotic Map. 6th International Multi-Conference on Systems, Signals and Devices, 2009, SSD ’09, 23-26 March 2009, 1-6.
http://dx.doi.org/10.1109/SSD.2009.4956666 [113] Yang, H.Q., Wong, K.-W., Liao, X.F., Zhang, W. and Wei, P.C. (2010) A Fast Image Encryption and Authentication Scheme Based on Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 3507-3517.
http://dx.doi.org/10.1016/j.cnsns.2010.01.004 [114] Yang, H.Q., Wong, K.-W., Liao, X.F., Wang, Y. and Yang, D.G. (2009) One-Way Hash Function Construction Based on Chaotic Map Network. Chaos, Solitons & Fractals, 41, 2566-2574.
http://dx.doi.org/10.1016/j.chaos.2008.09.056 [115] Lian, S., Sun, J. and Wang, Z. (2005) A Block Cipher Based on a Suitable Use of the Chaotic Standard Map. Chaos, Solitons & Fractals, 26, 117-129.
http://dx.doi.org/10.1016/j.chaos.2004.11.096 [116] Li, P., Li, Z., Halang, W.A. and Chen, G.R. (2007) A Stream Cipher Based on a Spatiotemporal Chaotic System. Chaos, Solitons & Fractals, 32, 1867-1876.
http://dx.doi.org/10.1016/j.chaos.2005.12.021 [117] Ali-Pacha, A., Hadj-Said, N., M’Hamed, A. and Belgoraf, A. (2007) Lorenz’s Attractor Applied to the Stream Cipher (Ali-Pacha Generator). Chaos, Solitons & Fractals, 33, 1762-1766.
http://dx.doi.org/10.1016/j.chaos.2006.03.009 [118] Tong, X. and Cui, M. (2008) Image Encryption with Compound Chaotic Sequence Cipher Shifting Dynamically. Image and Vision Computing, 26, 843-850.
http://dx.doi.org/10.1016/j.imavis.2007.09.005 [119] Gao, H., Zhang, Y.S., Liang, S.Y. and Li, D.Q. (2006) A New Chaotic Algorithm for Image Encryption. Chaos, Solitons & Fractals, 29, 393-399.
http://dx.doi.org/10.1016/j.chaos.2005.08.110 [120] Chee, C.Y. and Xu, D. (2006) Chaotic Encryption Using Discrete-Time Synchronous Chaos. Physics Letters A, 348, 284-292.
http://dx.doi.org/10.1016/j.physleta.2005.08.082 [121] Gao, T. and Chen, Z. (2008) A New Image Encryption Algorithm Based on Hyper-Chaos. Physics Letters A, 372, 394-400.
http://dx.doi.org/10.1016/j.physleta.2007.07.040 [122] Sun, F. and Liu, S. (2009) Cryptographic Pseudo-Random Sequence from the Spatial Chaotic Map. Chaos, Solitons & Fractals, 41, 2216-2219.
http://dx.doi.org/10.1016/j.chaos.2008.08.032 [123] Li, P., Li, Z., Halang, W.A. and Chen, G.R. (2006) A Multiple Pseudorandom-Bit Generator Based on a Spatiotemporal Chaotic Map. Physics Letters A, 349, 467-473.
http://dx.doi.org/10.1016/j.physleta.2005.09.060 [124] Arroyo, D., Li, C.Q., Li, S.J. and Alvarez, G. (2009) Cryptanalysis of a Computer Cryptography Scheme Based on a Filter Bank. Chaos, Solitons & Fractals, 41, 410-413.
http://dx.doi.org/10.1016/j.chaos.2008.01.020 [125] Kanso, A. and Smaoui, N. (2009) Logistic Chaotic Maps for Binary Numbers Generations. Chaos, Solitons & Fractals, 40, 2557-2568.
http://dx.doi.org/10.1016/j.chaos.2007.10.049
[1] Sneyers, R. (1997) Climate Chaotic Instability: Statistical Determination and Theoretical Background. Environmetrics, 8, 517-532.
http://dx.doi.org/10.1002/(SICI)1099-095X(199709/10)8:5<517::AID-ENV267>3.0.CO;2-L [2] Zeng, X., Pielke, R.A. and Eykholt, R. (1993) Chaos Theory and Its Application to the Atmosphere. Bulletin of the American Meteorological Society, 74, 631-639.
http://dx.doi.org/10.1175/1520-0477(1993)074<0631:CTAIAT>2.0.CO;2 [3] Wikipedia. Chaos Theory. Cited 17 January 2009.
http://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=264934743 [4] Serletis, A. and Gogas, P. (2000) Purchasing Power Parity, Nonlinearity and Chaos. Applied Financial Economics, 10, 615-622.
http://dx.doi.org/10.1080/096031000437962 [5] Serletis, A. and Gogas, P. (1997) Chaos in East European Black Market Exchange Rates. Research in Economics, 51, 359-385.
http://dx.doi.org/10.1006/reec.1997.0050 [6] Wikipedia (2015) Chaos Theory.
https://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=693847517 [7] Maqableh, M., Samsudin, A.B. and Alia, M.A. (2008) New Hash Function Based on Chaos Theory (CHA-1). IJCSNS International Journal of Computer Science and Network Security, 8, 20-26. [8] Kocarev, L. (2001) Chaos-Based Cryptography: A Brief Overview. IEEE Circuits and Systems Magazine, 1, 6-21.
http://dx.doi.org/10.1109/7384.963463 [9] Pareek, N.K., Patidar, V. and Sud, K.K. (2003) Discrete Chaotic Cryptography Using External Key. Physics Letters A, 309, 75-82.
http://dx.doi.org/10.1016/S0375-9601(03)00122-1 [10] Pareek, N.K., Patidar, V. and Sud, K.K. (2005) Cryptography Using Multiple One-Dimensional Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 10, 715-723.
http://dx.doi.org/10.1016/j.cnsns.2004.03.006 [11] Xiang, T., Liao, X.F., Tang, G.P., Chen, Y. and Wong, K.W. (2006) A Novel Block Cryptosystem Based on Iterating a Chaotic Map. Physics Letters A, 349, 109-115.
http://dx.doi.org/10.1016/j.physleta.2005.02.083 [12] Chen, S., Zhong, X.X. and Wu, Z.Z. (2008) Chaos Block Cipher for Wireless Sensor Network. Science in China Series F: Information Sciences, 51, 1055-1063.
http://dx.doi.org/10.1007/s11432-008-0102-5 [13] Peng, J., You, M.Y., Yang, Z.M. and Jin, S.Z. (2007) Research on a Block Encryption Cipher Based on Chaotic Dynamical System. 3rd International Conference on Natural Computation, ICNC 2007, Haikou, 24-27 August 2007, 744-748.
http://dx.doi.org/10.1109/ICNC.2007.612 [14] Yang, H.Q., Liao, X.F., Wong, K.W., Zhang, W. and Wei, P.C. (2007) A New Block Cipher Based on Chaotic Map and Group Theory. Chaos, Solitons & Fractals, 40, 50-59.
http://dx.doi.org/10.1016/j.chaos.2007.07.056 [15] Peng, J., Jin, S.Z., Chen, G.R., Yang, Z.M. and Liao, X.F. (2008) An Image Encryption Scheme Based on Chaotic Map. 4th International Conference on Natural Computation, ICNC ’08, Jinan, 18-20 October 2008, 595-599.
http://dx.doi.org/10.1109/icnc.2008.227 [16] Lian, S. (2009) A Block Cipher Based on Chaotic Neural Networks. Neurocomputing, 72, 1296-1301.
http://dx.doi.org/10.1016/j.neucom.2008.11.005 [17] Wang, F.J., Zhang, Y.P. and Cao, T.J. (2009) Research of Chaotic Block Cipher Algorithm Based on Logistic Map. 2nd International Conference on Intelligent Computation Technology and Automation, 2009, ICICTA ’09, Changsha, 10-11 October 2009, 678-681.
http://dx.doi.org/10.1109/ICICTA.2009.169 [18] Peng, J., Jin, S.Z., Liu, H.L. and Liu, Y.G. (2009) A Block Cipher Based on a Hybrid of Chaotic System and Feistel Network. 5th International Conference on Natural Computation, ICNC ’09, Tianjin, 14-16 August 2009, 427-431.
http://dx.doi.org/10.1109/icnc.2009.663 [19] Amin, M., Faragallah, O.S. and Abd El-Latif, A.A. (2010) A Chaotic Block Cipher Algorithm for Image Cryptosystems. Communications in Nonlinear Science and Numerical Simulation, 15, 3484-3497.
http://dx.doi.org/10.1016/j.cnsns.2009.12.025 [20] Huang, J.-H. and Liu, Y. (2010) A Block Encryption Algorithm Combined with the Logistic Mapping and SPN Structure. 2010 2nd International Conference on Industrial and Information Systems (IIS), Dalian, 10-11 July 2010, 156-159.
http://dx.doi.org/10.1109/indusis.2010.5565655 [21] Zhao, G., Chen, G.R., Fang, J.Q. and Xu, G. (2011) Block Cipher Design: Generalized Single-Use-Algorithm Based on Chaos. Tsinghua Science & Technology, 16, 194-206.
http://dx.doi.org/10.1016/S1007-0214(11)70030-X [22] Masuda, N., Jakimoski, G., Aihara, K. and Kocarev, L. (2006) Chaotic Block Ciphers: From Theory to Practical Algorithms. IEEE Transactions on Circuits and Systems I: Regular Papers, 53, 1341-1352.
http://dx.doi.org/10.1109/TCSI.2006.874182 [23] Habutsu, T., Nishio, Y., Sasase, I. and Mori, S. (1991) A Secret Key Cryptosystem by Iterating a Chaotic Map. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 127-140.
http://dx.doi.org/10.1007/3-540-46416-6_11 [24] Gutowitz, H.A. (1993) Cryptography with Dynamical Systems, in Cellular Automata and Cooperative Phenomena. Kluwer Academic Press, Dordrecht.
http://dx.doi.org/10.1007/978-94-011-1691-6_21 [25] Kotulski, Z. and Szczepanski, J. (1997) Discrete Chaotic Cryptography (DCC). Annalen der Physik, 6, 381-394.
http://dx.doi.org/10.1002/andp.19975090504 [26] García, P. and Jiménez, J. (2002) Communication through Chaotic Map Systems. Physics Letters A, 298, 35-40.
http://dx.doi.org/10.1016/S0375-9601(02)00382-1 [27] Masuda, N. and Aihara, K. (2002) Cryptosystems with Discretized Chaotic Maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49, 28-40.
http://dx.doi.org/10.1109/81.974872 [28] Baptista, M.S. (1998) Cryptography with Chaos. Physics Letters A, 240, 50-54.
http://dx.doi.org/10.1016/S0375-9601(98)00086-3 [29] Wong, W.-K., Lee, L.-P. and Wong, K.-W. (2001) A Modified Chaotic Cryptographic Method. Computer Physics Communications, 138, 234-236.
http://dx.doi.org/10.1016/S0010-4655(01)00220-X [30] Alvarez, E., Fernández, A., Garcia, P., Jiménez, J. and Marcano, A. (1999) New Approach to Chaotic Encryption. Physics Letters A, 263, 373-375.
http://dx.doi.org/10.1016/S0375-9601(99)00747-1 [31] Wong, K.W. (2002) A Fast Chaotic Cryptographic Scheme with Dynamic Look-Up Table. Physics Letters A, 298, 238-242.
http://dx.doi.org/10.1016/S0375-9601(02)00431-0 [32] Machado, R.F., Baptista, M.S. and Grebogi, C. (2004) Cryptography with Chaos at the Physical Level. Chaos, Solitons & Fractals, 21, 1265-1269.
http://dx.doi.org/10.1016/j.chaos.2003.12.094 [33] Guan, Z.-H., Huang, F. and Guan, W. (2005) Chaos-Based Image Encryption Algorithm. Physics Letters A, 346, 153-157.
http://dx.doi.org/10.1016/j.physleta.2005.08.006 [34] Gao, T. and Chen, Z. (2008) Image Encryption Based on a New Total Shuffling Algorithm. Chaos, Solitons & Fractals, 38, 213-220.
http://dx.doi.org/10.1016/j.chaos.2006.11.009 [35] Wong, K.W. (2003) A Combined Chaotic Cryptographic and Hashing Scheme. Physics Letters A, 307, 292-298.
http://dx.doi.org/10.1016/S0375-9601(02)01770-X [36] Xiao, D., Liao, X. and Deng, S. (2005) One-Way Hash Function Construction Based on the Chaotic Map with Changeable-Parameter. Chaos, Solitons & Fractals, 24, 65-71. http://dx.doi.org/10.1016/S0960-0779(04)00456-4 [37] Lian, S.G., Liu, Z.X., Ren, Z. and Wang, H.L. (2006) Hash Function Based on Chaotic Neural Networks. 2006 IEEE International Symposium on Circuits and Systems, ISCAS 2006, Island of Kos, 21-24 May 2006.
http://dx.doi.org/10.1109/iscas.2006.1692566 [38] Peng, F. and Qiu, S.-S. (2007) One-Way Hash Functions Based on Iterated Chaotic Systems. International Conference on Communications, Circuits and Systems, ICCCAS 2007, Kokura, 11-13 July 2007, 1070-1074.
http://dx.doi.org/10.1109/ICCCAS.2007.4348231 [39] Khan, M.K., Zhang, J. and Wang, X. (2008) Chaotic Hash-Based Fingerprint Biometric Remote User Authentication Scheme on Mobile Devices. Chaos, Solitons & Fractals, 35, 519-524.
http://dx.doi.org/10.1016/j.chaos.2006.05.061 [40] Xiao, D., Liao, X. and Deng, S. (2008) Parallel Keyed Hash Function Construction Based on Chaotic Maps. Physics Letters A, 372, 4682-4688.
http://dx.doi.org/10.1016/j.physleta.2008.04.060 [41] Song, Y.R. and Jiang, G.P. (2008) Hash Function Construction Based on Chaotic Coupled Map Network. The 9th International Conference for Young Computer Scientists, ICYCS 2008, Hunan, 18-21 November 2008, 2753-2758.
http://dx.doi.org/10.1109/ICYCS.2008.134 [42] Deng, S., Xiao, D., Li, Y.T. and Peng, W.B. (2009) A Novel Combined Cryptographic and Hash Algorithm Based on Chaotic Control Character. Communications in Nonlinear Science and Numerical Simulation, 14, 3889-3900.
http://dx.doi.org/10.1016/j.cnsns.2009.02.020 [43] Guyeux, C. and Bahi, J.M. (2010) Topological Chaos and Chaotic Iterations Application to Hash Functions. The 2010 International Joint Conference on Neural Networks (IJCNN), Barcelona, 18-23 July 2010, 1-7.
http://dx.doi.org/10.1109/ijcnn.2010.5596512 [44] Huang, Z. (2011) A More Secure Parallel Keyed Hash Function Based on Chaotic Neural Network. Communications in Nonlinear Science and Numerical Simulation, 16, 3245-3256. http://dx.doi.org/10.1016/j.cnsns.2010.12.009 [45] Wang, Y., Wong, K.-W. and Xiao, D. (2011) Parallel Hash Function Construction Based on Coupled Map Lattices. Communications in Nonlinear Science and Numerical Simulation, 16, 2810-2821.
http://dx.doi.org/10.1016/j.cnsns.2010.10.001 [46] Yi, X. (2005) Hash Function Based on Chaotic Tent Maps. IEEE Transactions on Circuits and Systems II: Express Briefs, 52, 354-357.
http://dx.doi.org/10.1109/TCSII.2005.848992 [47] Zhang, J., Wang, X. and Zhang, W. (2007) Chaotic Keyed Hash Function Based on Feedforward-Feedback Nonlinear Digital Filter. Physics Letters A, 362, 439-448.
http://dx.doi.org/10.1016/j.physleta.2006.10.052 [48] Amin, M., Faragallah, O.S. and Abd El-Latif, A.A. (2009) Chaos-Based Hash Function (CBHF) for Cryptographic Applications. Chaos, Solitons & Fractals, 42, 767-772.
http://dx.doi.org/10.1016/j.chaos.2009.02.001 [49] Xiao, D., Shih, F.Y. and Liao, X. (2010) A Chaos-Based Hash Function with both Modification Detection and Localization Capabilities. Communications in Nonlinear Science and Numerical Simulation, 15, 2254-2261.
http://dx.doi.org/10.1016/j.cnsns.2009.10.012 [50] Kocarev, L. and Jakimoski, G. (2003) Pseudorandom Bits Generated by Chaotic Maps. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50, 123-126.
http://dx.doi.org/10.1109/TCSI.2002.804550 [51] Tong, X.-J., Cui, M.-G. and Jiang, W. (2006) The Production Algorithm of Pseudo-Random Number Generator Based on Compound Non-Linear Chaos System. International Conference on Intelligent Information Hiding and Multimedia Signal Processing, 2006, IIH-MSP ’06, Pasadena, 18-20 December 2006, 685-688.
http://dx.doi.org/10.1109/IIH-MSP.2006.265094 [52] Chen, S. and Zhong, X.-X. (2007) Chaotic Block Iterating Method for Pseudo-Random Sequence Generator. The Journal of China Universities of Posts and Telecommunications, 14, 45-48.
http://dx.doi.org/10.1016/s1005-8885(07)60054-5 [53] Zheng, F., Tian, X.-J., Song, J.-Y. and Li, X.-Y. (2008) Pseudo-Random Sequence Generator Based on the Generalized Henon Map. The Journal of China Universities of Posts and Telecommunications, 15, 64-68.
http://dx.doi.org/10.1016/S1005-8885(08)60109-0 [54] Qi, A.X., Han, C.Y. and Wang, G.Y. (2010) Design and FPGA Realization of a Pseudo Random Sequence Generator Based on a Switched Chaos. International Conference on Communications, Circuits and Systems (ICCCAS), Chengdu, 28-30 July 2010, 417-420.
http://dx.doi.org/10.1109/ICCCAS.2010.5581965 [55] Yoon, J.W. and Kim, H. (2010) An Image Encryption Scheme with a Pseudorandom Permutation Based on Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 3998-4006.
http://dx.doi.org/10.1016/j.cnsns.2010.01.041 [56] Dabal, P. and Pelka, R. (2011) A Chaos-Based Pseudo-Random Bit Generator Implemented in FPGA Device. 2011 IEEE 14th International Symposium on Design and Diagnostics of Electronic Circuits & Systems (DDECS), Cottbus, 13-15 April 2011, 151-154.
http://dx.doi.org/10.1109/ddecs.2011.5783069 [57] Forré, R. (1991) The Hénon Attractor as a Keystream Generator. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 76-81. [58] Matthews, R.A.J. (1989) On the Derivation of a “Chaotic” Encryption Algorithm. Cryptologia, 13, 29-42.
http://dx.doi.org/10.1080/0161-118991863745 [59] Li, S., Mou, X. and Cai, Y. (2001) Improving Security of a Chaotic Encryption Approach. Physics Letters A, 290, 127-133.
http://dx.doi.org/10.1016/S0375-9601(01)00612-0 [60] Wolfram, S. (1985) Cryptography with Cellular Automata. In: Williams, H.C., Ed., Advances in Cryptology— CRYPTO ’85 Proceedings, Lecture Notes in Computer Science, Spinger-Verlag, Berlin, 429-432. [61] Lee, P.-H., Pei, S.-C. and Chen, Y.-Y. (2003) Generating Chaotic Stream Ciphers Using Chaotic Systems. Chinese Journal of Physics, 41, 559-581. [62] Sang, T., Wang, R. and Yan, Y. (2000) Constructing Chaotic Discrete Sequences for Digital Communications Based on Correlation Analysis. IEEE Transactions on Signal Processing, 48, 2557-2565.
http://dx.doi.org/10.1109/78.863058 [63] Kwok, H.S. and Tang, W.K.S. (2007) A Fast Image Encryption System Based on Chaotic Maps with Finite Precision Representation. Chaos, Solitons & Fractals, 32, 1518-1529.
http://dx.doi.org/10.1016/j.chaos.2005.11.090 [64] Patidar, V., Pareek, N.K. and Sud, K.K. (2009) A New Substitution-Diffusion Based Image Cipher Using Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 14, 3056-3075.
http://dx.doi.org/10.1016/j.cnsns.2008.11.005 [65] Patidar, V., Pareek, N.K., Purohit, G. and Sud, K.K. (2010) Modified Substitution-Diffusion Image Cipher Using Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 2755-2765.
http://dx.doi.org/10.1016/j.cnsns.2009.11.010 [66] Biham, E. (1991) Cryptanalysis of the Chaotic-Map Cryptosystem Suggested at EUROCRYPT’91. In: Davies, D.W., Ed., Advances in Cryptology—EUROCRYPT ’91, Springer, Berlin, 532-534.
http://dx.doi.org/10.1007/3-540-46416-6_49 [67] Wheeler, D.D. (1989) Problems with Chaotic Cryptosystems. Cryptologia, 13, 243-250.
http://dx.doi.org/10.1080/0161-118991863934 [68] Alvarez, G., Montoya, F., Romera, M. and Pastor, G. (2000) Cryptanalysis of a Chaotic Encryption System. Physics Letters A, 276, 191-196.
http://dx.doi.org/10.1016/S0375-9601(00)00642-3 [69] Alvarez, G., Montoya, F., Romera, M. and Pastor, G. (2003) Cryptanalysis of a Discrete Chaotic Cryptosystem Using External Key. Physics Letters A, 319, 334-339.
http://dx.doi.org/10.1016/j.physleta.2003.10.044 [70] Wei, J., Liao, X.F., Wong, K.W. and Zhou, T. (2007) Cryptanalysis of a Cryptosystem Using Multiple One-Dimensional Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 12, 814-822.
http://dx.doi.org/10.1016/j.cnsns.2005.06.001 [71] Li, C.Q., Li, S.J., Alvarez, G., Chen, G.R. and Lo, K.-T. (2008) Cryptanalysis of a Chaotic Block Cipher with External Key and Its Improved Version. Chaos, Solitons & Fractals, 37, 299-307.
http://dx.doi.org/10.1016/j.chaos.2006.08.025 [72] Wang, X. and Yu, C. (2009) Cryptanalysis and Improvement on a Cryptosystem Based on a Chaotic Map. Computers & Mathematics with Applications, 57, 476-482.
http://dx.doi.org/10.1016/j.camwa.2008.09.042 [73] Yang, J., Xiao, D. and Xiang, T. (2011) Cryptanalysis of a Chaos Block Cipher for Wireless Sensor Network. Communications in Nonlinear Science and Numerical Simulation, 16, 844-850.
http://dx.doi.org/10.1016/j.cnsns.2010.05.005 [74] álvarez, G., Montoya, F., Romera, M. and Pastor, G. (2004) Keystream Cryptanalysis of a Chaotic Cryptographic Method. Computer Physics Communications, 156, 205-207.
http://dx.doi.org/10.1016/S0010-4655(03)00432-6 [75] Jakimoski, G. and Kocarev, L. (2001) Analysis of Some Recently Proposed Chaos-Based Encryption Algorithms. Physics Letters A, 291, 381-384.
http://dx.doi.org/10.1016/S0375-9601(01)00771-X [76] Li, S.J., Mou, X.Q., Ji, Z., Zhang, J.H. and Cai, Y.L. (2003) Performance Analysis of Jakimoski-Kocarev Attack on a Class of Chaotic Cryptosystems. Physics Letters A, 307, 22-28.
http://dx.doi.org/10.1016/S0375-9601(02)01659-6 [77] ?okal, C. and Solak, E. (2009) Cryptanalysis of a Chaos-Based Image Encryption Algorithm. Physics Letters A, 373, 1357-1360.
http://dx.doi.org/10.1016/j.physleta.2009.02.030 [78] Maqableh, M.M. and Dantchev, S. (2009) Cryptanalysis of Chaos-Based Hash Function (CBHF). 1st International Alternative Workshop on Aggressive Computing and Security—iAWACS, Laval, 23-25 October 2009, 20-30. [79] Yang, Q.-T., Gao, T.-G., Fan, L. and Gu, Q.-L. (2009) Analysis of One-Way Alterable Length Hash Function Based on Cell Neural Network. 5th International Conference on Information Assurance and Security, IAS ’09, Xi’an, 18-20 August 2009, 391-395.
http://dx.doi.org/10.1109/ias.2009.87 [80] Xiao, D., Liao, X. and Wang, Y. (2009) Improving the Security of a Parallel Keyed Hash Function Based on Chaotic Maps. Physics Letters A, 373, 4346-4353.
http://dx.doi.org/10.1016/j.physleta.2009.09.059 [81] Deng, S., Li, Y. and Xiao, D. (2010) Analysis and Improvement of a Chaos-Based Hash Function Construction. Communications in Nonlinear Science and Numerical Simulation, 15, 1338-1347.
http://dx.doi.org/10.1016/j.cnsns.2009.05.065 [82] Li, C.Q., Li, S.J., Chen, G.R. and Halang, W.A. (2009) Cryptanalysis of an Image Encryption Scheme Based on a Compound Chaotic Sequence. Image and Vision Computing, 27, 1035-1039.
http://dx.doi.org/10.1016/j.imavis.2008.09.004 [83] Rhouma, R., Solak, E. and Belghith, S. (2010) Cryptanalysis of a New Substitution-Diffusion Based Image Cipher. Communications in Nonlinear Science and Numerical Simulation, 15, 1887-1892.
http://dx.doi.org/10.1016/j.cnsns.2009.07.007 [84] Li, C., Li, S. and Lo, K.-T. (2011) Breaking a Modified Substitution-Diffusion Image Cipher Based on Chaotic Standard and Logistic Maps. Communications in Nonlinear Science and Numerical Simulation, 16, 837-843.
http://dx.doi.org/10.1016/j.cnsns.2010.05.008 [85] Maqableh, M.M. (2011) Fast Parallel Keyed Hash Functions Based on Chaotic Maps (PKHC). Western European Workshop on Research in Cryptology, Weimar, 20-22 July 2011, 33-40. [86] Maqableh, M.M. (2010) Secure Hash Functions Based on Chaotic Maps for E-Commerce Application. International Journal of Information Technology and Management information System (IJITMIS), 1, 12-19. [87] Maqableh, M.M. (2010) Fast Hash Function Based on BCCM Encryption Algorithm for E-Commerce (HFBCCM). 5th International Conference on e-Commerce in Developing Countries: With Focus on Export, Le Havre, 15-16 September 2010, 55-64. [88] Bertuglia, C.S. and Vaio, F. (2005) Nonlinearity, Chaos & Complexity: The Dynamics of Natural and Social Systems. Oxford University Press. [89] Alligood, K.T., Sauer, T.D. and Yorke, J.A. (1996) Chaos an Introduction to Dynamical Systems. Springer-Verlag, New York. [90] Solari, H.G., Natiello, M.A. and Mindlin, G.B. (1996) Nonlinear Dynamics A Two-Way Trip from Physics to Math. Institute of Physics Publishing, Bristol. [91] Baker, G.L. and Gollub, J.P. (1990) Chaotic Dynamics an Introduction. Press Syndicate of the University of Cambridge, New York. [92] Poincaré, J.H. (1890) Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt. Acta Mathematica, 13, 1-270. [93] Lorenz, E.N. (1963) Deterministic Nonperiodic Flow. Journal of Atmospheric Sciences, 20, 130-141.
http://dx.doi.org/10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2 [94] Pritchard, J. (1996) The Chaos Cookbook. Butterworh-Heinemann, Oxford. [95] Rahimi, A., Mohammadi, S. and Rahimi, R. (2009) An Efficient Iris Authentication Using Chaos Theory-Based Cryptography for E-Commerce Transactions. International Conference for Internet Technology and Secured Transactions (ICITST 2009). [96] Serletis, A. and Gogas, P. (1999) The North American Natural Gas Liquids Markets Are Chaotic. The Energy Journal, 20, 83-103.
http://dx.doi.org/10.5547/ISSN0195-6574-EJ-Vol20-No1-5 [97] Gilmore, R. (2004) Chaos and Attractors. Encyclopedia of Mathematical Physics, EMP MS 93. [98] Tullaro, N.B., Abbott, T. and Reilly, J.P. (1992) An Experimental Approach to Nonlinear Dynamics and Chaos. Vol. 1, Addison-Wesley, Boston. [99] Parker, T.S. and Chua, L.O. (1989) Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-3486-9 [100] Schmitz, R. (2001) Use of Chaotic Dynamical Systems in Cryptography. Journal of the Franklin Institute, 338, 429-441.
http://dx.doi.org/10.1016/S0016-0032(00)00087-9 [101] Maqableh, M.M. (2012) Analysis and Design Security Primitives Based on Chaotic Systems for eCommerce. Doctoral Thesis, Durham University, Durham. [102] Grassi, G. and Mascoio, S. (1999) Synchronizing Hyperchaotic Systems by Observer Design. IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 46, 478-483.
http://dx.doi.org/10.1109/82.755422 [103] May, R.M. (1976) Simple Mathematical Models with Very Complicated Dynamics. Nature, 261, 459-467.
http://dx.doi.org/10.1038/261459a0 [104] Wikipedia. Logistic Map. Cited 20 February 2009.
http://en.wikipedia.org/w/index.php?title=chaotic_maps&oldid=261864353 [105] Naess, A. (2000) Chaos and Nonlinear Stochastic Dynamics. Probabilistic Engineering Mechanics, 15, 37-47.
http://dx.doi.org/10.1016/S0266-8920(99)00007-7 [106] Contributors, W. (2005) Chaos Theory.
http://en.wikipedia.org/w/index.php?title=Chaos_theory&oldid=264934743 [107] Rossler, O. (1976) An Equation for Continous Chaos. Physics Letters A, 57, 397-398.
http://dx.doi.org/10.1016/0375-9601(76)90101-8 [108] Ho, A. (2006) Chaos Introduction.
http://www.zeuscat.com/andrew/chaos/chaos.html [109] Wikipedia. Tent Map. Cited 25 February 2009.
http://en.wikipedia.org/w/index.php?title=Tent_map&oldid=186656075 [110] Li, S., Chen, G. and Mou, X. (2005) On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps. International Journal of Bifurcation and Chaos, 15, 3119-3151. http://dx.doi.org/10.1142/S0218127405014052 [111] Jun, P., Jin, S.Z., Liu, Y.G., Yang, Z.M., You, M.Y. and Pei, Y.J. (2008) A Novel Scheme for Image Encryption Based on Piecewise Linear Chaotic Map. 2008 IEEE Conference on Cybernetics and Intelligent Systems, Chengdu, 21-24 September 2008, 1012-1016. [112] Rhouma, R., Arroyo, D. and Belghith, S. (2009) A New Color Image Cryptosystem Based on a Piecewise Linear Chaotic Map. 6th International Multi-Conference on Systems, Signals and Devices, 2009, SSD ’09, 23-26 March 2009, 1-6.
http://dx.doi.org/10.1109/SSD.2009.4956666 [113] Yang, H.Q., Wong, K.-W., Liao, X.F., Zhang, W. and Wei, P.C. (2010) A Fast Image Encryption and Authentication Scheme Based on Chaotic Maps. Communications in Nonlinear Science and Numerical Simulation, 15, 3507-3517.
http://dx.doi.org/10.1016/j.cnsns.2010.01.004 [114] Yang, H.Q., Wong, K.-W., Liao, X.F., Wang, Y. and Yang, D.G. (2009) One-Way Hash Function Construction Based on Chaotic Map Network. Chaos, Solitons & Fractals, 41, 2566-2574.
http://dx.doi.org/10.1016/j.chaos.2008.09.056 [115] Lian, S., Sun, J. and Wang, Z. (2005) A Block Cipher Based on a Suitable Use of the Chaotic Standard Map. Chaos, Solitons & Fractals, 26, 117-129.
http://dx.doi.org/10.1016/j.chaos.2004.11.096 [116] Li, P., Li, Z., Halang, W.A. and Chen, G.R. (2007) A Stream Cipher Based on a Spatiotemporal Chaotic System. Chaos, Solitons & Fractals, 32, 1867-1876.
http://dx.doi.org/10.1016/j.chaos.2005.12.021 [117] Ali-Pacha, A., Hadj-Said, N., M’Hamed, A. and Belgoraf, A. (2007) Lorenz’s Attractor Applied to the Stream Cipher (Ali-Pacha Generator). Chaos, Solitons & Fractals, 33, 1762-1766.
http://dx.doi.org/10.1016/j.chaos.2006.03.009 [118] Tong, X. and Cui, M. (2008) Image Encryption with Compound Chaotic Sequence Cipher Shifting Dynamically. Image and Vision Computing, 26, 843-850.
http://dx.doi.org/10.1016/j.imavis.2007.09.005 [119] Gao, H., Zhang, Y.S., Liang, S.Y. and Li, D.Q. (2006) A New Chaotic Algorithm for Image Encryption. Chaos, Solitons & Fractals, 29, 393-399.
http://dx.doi.org/10.1016/j.chaos.2005.08.110 [120] Chee, C.Y. and Xu, D. (2006) Chaotic Encryption Using Discrete-Time Synchronous Chaos. Physics Letters A, 348, 284-292.
http://dx.doi.org/10.1016/j.physleta.2005.08.082 [121] Gao, T. and Chen, Z. (2008) A New Image Encryption Algorithm Based on Hyper-Chaos. Physics Letters A, 372, 394-400.
http://dx.doi.org/10.1016/j.physleta.2007.07.040 [122] Sun, F. and Liu, S. (2009) Cryptographic Pseudo-Random Sequence from the Spatial Chaotic Map. Chaos, Solitons & Fractals, 41, 2216-2219.
http://dx.doi.org/10.1016/j.chaos.2008.08.032 [123] Li, P., Li, Z., Halang, W.A. and Chen, G.R. (2006) A Multiple Pseudorandom-Bit Generator Based on a Spatiotemporal Chaotic Map. Physics Letters A, 349, 467-473.
http://dx.doi.org/10.1016/j.physleta.2005.09.060 [124] Arroyo, D., Li, C.Q., Li, S.J. and Alvarez, G. (2009) Cryptanalysis of a Computer Cryptography Scheme Based on a Filter Bank. Chaos, Solitons & Fractals, 41, 410-413.
http://dx.doi.org/10.1016/j.chaos.2008.01.020 [125] Kanso, A. and Smaoui, N. (2009) Logistic Chaotic Maps for Binary Numbers Generations. Chaos, Solitons & Fractals, 40, 2557-2568.
http://dx.doi.org/10.1016/j.chaos.2007.10.049