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Askar SS, Karawia AA, Al-Khedhairi A, Al-Ammar FS. An Algorithm of Image Encryption Using Logistic and Two-Dimensional Chaotic Economic Maps. ENTROPY 2019; 21:e21010044. [PMID: 33266760 PMCID: PMC7514150 DOI: 10.3390/e21010044] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/26/2018] [Revised: 12/30/2018] [Accepted: 12/31/2018] [Indexed: 11/24/2022]
Abstract
In the literature, there are many image encryption algorithms that have been constructed based on different chaotic maps. However, those algorithms do well in the cryptographic process, but still, some developments need to be made in order to enhance the security level supported by them. This paper introduces a new cryptographic algorithm that depends on a logistic and two-dimensional chaotic economic map. The robustness of the introduced algorithm is shown by implementing it on several types of images. The implementation of the algorithm and its security are partially analyzed using some statistical analyses such as sensitivity to the key space, pixels correlation, the entropy process, and contrast analysis. The results given in this paper and the comparisons performed have led us to decide that the introduced algorithm is characterized by a large space of key security, sensitivity to the secret key, few coefficients of correlation, a high contrast, and accepted information of entropy. In addition, the results obtained in experiments show that our proposed algorithm resists statistical, differential, brute-force, and noise attacks.
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Affiliation(s)
- Sameh S Askar
- Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
- Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
| | - Abdel A Karawia
- Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
- Computer Science Unit, Deanship of Educational Services, Qassim University, P.O. Box 6595, Buraidah 51452, Saudi Arabia
| | - Abdulrahman Al-Khedhairi
- Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
| | - Fatemah S Al-Ammar
- Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
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Antonik P, Gulina M, Pauwels J, Massar S. Using a reservoir computer to learn chaotic attractors, with applications to chaos synchronization and cryptography. Phys Rev E 2018; 98:012215. [PMID: 30110744 DOI: 10.1103/physreve.98.012215] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2018] [Indexed: 06/08/2023]
Abstract
Using the machine learning approach known as reservoir computing, it is possible to train one dynamical system to emulate another. We show that such trained reservoir computers reproduce the properties of the attractor of the chaotic system sufficiently well to exhibit chaos synchronization. That is, the trained reservoir computer, weakly driven by the chaotic system, will synchronize with the chaotic system. Conversely, the chaotic system, weakly driven by a trained reservoir computer, will synchronize with the reservoir computer. We illustrate this behavior on the Mackey-Glass and Lorenz systems. We then show that trained reservoir computers can be used to crack chaos based cryptography and illustrate this on a chaos cryptosystem based on the Mackey-Glass system. We conclude by discussing why reservoir computers are so good at emulating chaotic systems.
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Affiliation(s)
- Piotr Antonik
- CentraleSupélec, Campus de Metz, Université Paris Saclay, F-57070 Metz, France
- LMOPS EA 4423 Laboratory, CentraleSupélec & Université de Lorraine, F-57070 Metz, France
| | - Marvyn Gulina
- Namur Institute for Complex Systems, Université de Namur, B-5000 Namur, Belgium
| | - Jaël Pauwels
- Applied Physics Research Group, Vrije Universiteit Brussels, B-1050 Brussels, Belgium
- Laboratoire d'Information Quantique, Université libre de Bruxelles, B-1050 Brussels, Belgium
| | - Serge Massar
- Laboratoire d'Information Quantique, Université libre de Bruxelles, B-1050 Brussels, Belgium
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3
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Chaos based crossover and mutation for securing DICOM image. Comput Biol Med 2016; 72:170-84. [DOI: 10.1016/j.compbiomed.2016.03.020] [Citation(s) in RCA: 143] [Impact Index Per Article: 17.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2015] [Revised: 03/10/2016] [Accepted: 03/24/2016] [Indexed: 11/23/2022]
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FRANK TD, FRIEDRICH R, BEEK PJ. TIME SERIES ANALYSIS OF MULTIVARIATE TIME-DELAYED SYSTEMS WITH NOISE: APPLICATIONS TO LASER PHYSICS AND HUMAN MOVEMENT. STOCH DYNAM 2011. [DOI: 10.1142/s0219493705001456] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
A data analysis method is proposed to reconstruct evolution equations of stochastic systems with time-delayed feedback from experimental time series. Multivariate systems involving noise sources with arbitrary correlation times are considered.
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Affiliation(s)
- T. D. FRANK
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - R. FRIEDRICH
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - P. J. BEEK
- Faculty of Human Movement Sciences, Vrije Universiteit, Van der Boechorststraat 9, 1081 BT Amsterdam, The Netherlands
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Ghosh D. Generalized projective synchronization in time-delayed systems: nonlinear observer approach. CHAOS (WOODBURY, N.Y.) 2009; 19:013102. [PMID: 19334966 DOI: 10.1063/1.3054711] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/27/2023]
Abstract
In this paper, we consider the projective-anticipating, projective, and projective-lag synchronization in a unified coupled time-delay system via nonlinear observer design. A new sufficient condition for generalized projective synchronization is derived analytically with the help of Krasovskii-Lyapunov theory for constant and variable time-delay systems. The analytical treatment can give stable synchronization (anticipatory and lag) for a large class of time-delayed systems in which the response system's trajectory is forced to have an amplitude proportional to the drive system. The constant of proportionality is determined by the control law, not by the initial conditions. The proposed technique has been applied to synchronize Ikeda and prototype models by numerical simulation.
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Affiliation(s)
- Dibakar Ghosh
- Department of Mathematics, Dinabandhu Andrews College, Garia, Calcutta, India.
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Ponomarenko VI, Prokhorov MD. Recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:066207. [PMID: 19256925 DOI: 10.1103/physreve.78.066207] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/11/2008] [Indexed: 05/27/2023]
Abstract
We propose a set of methods for the estimation of the parameters of time-delay systems with a linear filter and nonlinear delay feedback performing periodic oscillations. The methods are based on an analysis of the system response to regular external perturbations and are valid only for systems whose dynamics can be perturbed. The efficiency of the methods is illustrated using both numerical and experimental data.
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Affiliation(s)
- V I Ponomarenko
- Saratov Branch of the Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov, 410019, Russia
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Jinfeng H, Jingbo G. Breaking a chaotic secure communication scheme. CHAOS (WOODBURY, N.Y.) 2008; 18:013121. [PMID: 18377072 DOI: 10.1063/1.2885388] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/26/2023]
Abstract
The chaotic direct sequence spread spectrum (CD3S) has been extensively studied and accepted as an approach of chaotic secure communication. However, we find that CD3S is insecure and present the results of breaking CD3S. A modified unscented-Kalman-filter chaos fitting method is proposed. With the proposed method, an intruder can use a different chaotic system to fit the transmitter's chaotic system and break the CD3S scheme without any knowledge of the chaotic transmitter's structure, parameters or initial value, even under the following conditions: (A) the chaotic spreading sequence is complex and generated by two chaotic maps; (B) there exists noise in the CD3S signal; (C) the CD3S signal is constructed in frames. Simulation results verify the method.
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Affiliation(s)
- Hu Jinfeng
- Department of Electrical Engineering, Tsinghua University, Beijing 100084, People's Republic of China.
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Senthilkumar DV, Lakshmanan M. Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems. CHAOS (WOODBURY, N.Y.) 2007; 17:013112. [PMID: 17411248 DOI: 10.1063/1.2437651] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/14/2023]
Abstract
Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete, and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag, and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of the similarity functions and the intermittent behavior is characterized by a universal asymptotic -32 power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.
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Affiliation(s)
- D V Senthilkumar
- Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli-620 024, India.
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A New Image Encryption Algorithm Based on One-Dimensional Polynomial Chaotic Maps. COMPUTER AND INFORMATION SCIENCES – ISCIS 2006 2006. [DOI: 10.1007/11902140_100] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
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Frank TD. Delay Fokker-Planck equations, Novikov's theorem, and Boltzmann distributions as small delay approximations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:011112. [PMID: 16089942 DOI: 10.1103/physreve.72.011112] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2005] [Revised: 03/10/2005] [Indexed: 05/03/2023]
Abstract
We study time-delayed stochastic systems that can be described by means of so-called delay Fokker-Planck equations. Using Novikov's theorem, we first show that the theory of delay Fokker-Planck equations is on an equal footing with the theory of ordinary Fokker-Planck equations. Subsequently, we derive stationary distributions in the case of small time delays. In the case of additive noise systems, these distributions can be cast into the form of Boltzmann distributions involving effective potential functions.
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Affiliation(s)
- T D Frank
- Institute for Theoretical Physics, University of Münster, Germany
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Prokhorov MD, Ponomarenko VI. Estimation of coupling between time-delay systems from time series. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:016210. [PMID: 16090069 DOI: 10.1103/physreve.72.016210] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2004] [Revised: 01/03/2005] [Indexed: 05/03/2023]
Abstract
We propose a method for estimation of coupling between the systems governed by scalar time-delay differential equations of the Mackey-Glass type from the observed time series data. The method allows one to detect the presence of certain types of linear coupling between two time-delay systems, to define the type, strength, and direction of coupling, and to recover the model equations of coupled time-delay systems from chaotic time series corrupted by noise. We verify our method using both numerical and experimental data.
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Affiliation(s)
- M D Prokhorov
- Saratov Department of the Institute of RadioEngineering and Electronics of Russian Academy of Sciences, Zelyonaya Street, 38, Saratov, 410019, Russia
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Kye WH, Choi M, Kim CM, Park YJ. Encryption with synchronized time-delayed systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:045202. [PMID: 15903715 DOI: 10.1103/physreve.71.045202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2004] [Revised: 10/26/2004] [Indexed: 05/02/2023]
Abstract
We propose a new communication scheme that uses time-delayed chaotic systems with delay time modulation. In this method, the transmitter encodes a message as an additional modulation of the delay time and then the receiver decodes the message by tracking the delay time. We demonstrate our communication scheme in a system of coupled logistic maps. Also we discuss the error of the transferred message due to an external noise and present its correction method.
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Affiliation(s)
- Won-Ho Kye
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea
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Li X, Zhang H, Xue Y, Hu G. Enhancing chaoticity of spatiotemporal chaos. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:016216. [PMID: 15697707 DOI: 10.1103/physreve.71.016216] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2004] [Revised: 10/29/2004] [Indexed: 05/24/2023]
Abstract
In some practical situations strong chaos is needed. This introduces the task of chaos control with enhancing chaoticity rather than suppressing chaoticity. In this paper a simple method of linear amplifications incorporating modulo operations is suggested to make spatiotemporal systems, which may be originally chaotic or nonchaotic, strongly chaotic. Specifically, this control can eliminate periodic windows, increase the values and the number of positive Lyapunov exponents, make the probability distributions of the output chaotic sequences more homogeneous, and reduce the correlations of chaotic outputs for different times and different space units. The applicability of the method to practical tasks, in particular to random number generators and secure communications, is briefly discussed.
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Affiliation(s)
- Xiaowen Li
- Department of Physics, Beijing Normal University, Beijing 100875, People's Republic of China
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Ryu JW, Kye WH, Lee SY, Kim MW, Choi M, Rim S, Park YJ, Kim CM. Effects of time-delayed feedback on chaotic oscillators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:036220. [PMID: 15524625 DOI: 10.1103/physreve.70.036220] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2004] [Revised: 06/11/2004] [Indexed: 05/24/2023]
Abstract
We study the effects of time-delayed feedback on chaotic systems where the delay time is both fixed (static case) and varying (dynamic case) in time. For the static case, typical phase coherent and incoherent chaotic oscillators are investigated. Detailed phase diagrams are investigated in the parameter space of feedback gain ( K ) and delay time ( tau ). Linear stability analysis, by assuming the time-delayed perturbation, varies as e(lambdat) where lambda is the eigenvalue, gives the boundaries of the stability islands and critical feedback gains ( K(c) ) for both Rössler oscillators and Lorenz oscillators. We also found that the stability island are found when the delay time is about tau= (n+ 1 / 2 ) T , where n is an integer and T is the average period of the chaotic oscillator. It is shown that these analytical predictions agree well with the numerical results. For the dynamic case, we investigate Rössler oscillator with periodically modulated delay time. Stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time. We also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams.
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Affiliation(s)
- Jung-Wan Ryu
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea
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Kye WH, Choi M, Rim S, Kurdoglyan MS, Kim CM, Park YJ. Characteristics of a delayed system with time-dependent delay time. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:055202. [PMID: 15244871 DOI: 10.1103/physreve.69.055202] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/04/2003] [Indexed: 05/24/2023]
Abstract
The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate that the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.
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Affiliation(s)
- Won-Ho Kye
- National Creative Research Initiative Center for Controlling Optical Chaos, Pai-Chai University, Daejeon 302-735, Korea.
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Wang X, Zhan M, Lai CH, Gang H. Error function attack of chaos synchronization based encryption schemes. CHAOS (WOODBURY, N.Y.) 2004; 14:128-137. [PMID: 15003053 DOI: 10.1063/1.1633492] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Different chaos synchronization based encryption schemes are reviewed and compared from the practical point of view. As an efficient cryptanalysis tool for chaos encryption, a proposal based on the error function attack is presented systematically and used to evaluate system security. We define a quantitative measure (quality factor) of the effective applicability of a chaos encryption scheme, which takes into account the security, the encryption speed, and the robustness against channel noise. A comparison is made of several encryption schemes and it is found that a scheme based on one-way coupled chaotic map lattices performs outstandingly well, as judged from quality factor.
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Affiliation(s)
- Xingang Wang
- Temasek Laboratories, National University of Singapore, 119260, Singapore
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Chen JY, Wong KW, Cheng LM, Shuai JW. A secure communication scheme based on the phase synchronization of chaotic systems. CHAOS (WOODBURY, N.Y.) 2003; 13:508-514. [PMID: 12777114 DOI: 10.1063/1.1564934] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Phase synchronization of chaotic systems with both weak and strong couplings has recently been investigated extensively. Similar to complete synchronization, this type of synchronization can also be applied in secure communications. We develop a digital secure communication scheme that utilizes the instantaneous phase as the signal transmitted from the drive to the response subsystems. Simulation results show that the scheme is difficult to be broken by some traditional attacks. Moreover, it operates with a weak positive conditional Lyapunov exponent in the response subsystem.
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Affiliation(s)
- J Y Chen
- Department of Computer Engineering and Information Technology, City University of Hong Kong, Hong Kong, China
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