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  • Chaos Theory
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  • 1
    Language: English
    In: IEEE Transactions on Circuits and Systems-I: Fundamental Theory..., Jan, 2003, Vol.50(1), p.123(4)
    Description: Different possibilities of using chaotic maps as pseudorandom number generators are discussed. A class of chaos-based pseudorandom bit generators is constructed. We give simple arguments to explain the unpredictability of the generators. Index Terms--Chaos, pseudorandom numbers.
    Keywords: Chaos Theory ; Random Numbers ; Chaotic Systems
    ISSN: 1057-7122
    E-ISSN: 15581268
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  • 2
    Language: English
    In: IEEE Transactions on Circuits and Systems-I-Regular Papers, June, 2006, Vol.53(6), p.1300(10)
    Description: We propose a definition of the discrete Lyapunov exponent for an arbitrary permutation of a finite lattice. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by proving that, for large classes of chaotic maps, the corresponding discrete Lyapunov exponent approaches the largest Lyapunov exponent of a chaotic map when M [right arrow] [infinity], where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has discrete chaos if its discrete Lyapunov exponent tends to a positive number, when M [right arrow] [infinity]. We present several examples to illustrate the concepts being introduced. Index Terms--Chaos, discrete chaos, Lyapunov components.
    Keywords: Chaos Theory -- Analysis ; Liapunov Functions -- Usage ; Discrete Time Systems -- Usage
    ISSN: 1549-8328
    E-ISSN: 15580806
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  • 3
    In: IEEE Transactions on Circuits and Systems-I: Fundamental Theory..., Feb, 2001, Vol.48(2), p.163
    Description: This paper is devoted to the analysis of the impact of chaos-based techniques on block encryption ciphers. We present several chaos based ciphers. Using the well-known principles in the cryptanalysis we show that these ciphers do not behave worse than the standard ones, opening in this way a novel approach to the design of block encryption ciphers. Index Terms--Block encryption ciphers, chaos, cryptography, S-boxes.
    Keywords: Data Encryption -- Research ; Cryptography -- Research
    ISSN: 1057-7122
    E-ISSN: 15581268
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  • 4
    In: IEEE Transactions on Circuits and Systems-I: Fundamental Theory..., March, 2001, Vol.48(3), p.281
    Description: This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewise-linear one-dimensional(PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source, and then to mathematically analyze the information generation process and the RNG. In the companion paper we discuss practical aspects of our chaos-based RNGs. Index Terms--Chaos, random number generator, symbolic dynamics.
    Keywords: Random Number Generators (Software) -- Models ; Chaotic Systems -- Usage ; Markov Processes -- Analysis
    ISSN: 1057-7122
    E-ISSN: 15580806
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  • 5
    Language: English
    In: IEEE Transactions on Information Theory, April, 2006, Vol.52(4), p.1366(19)
    Description: Iterative decoding algorithms may be viewed as high-dimensional nonlinear dynamical systems, depending on a large number of parameters. In this work, we introduce a simplified description of several iterative decoding algorithms in terms of the a posteriori average entropy, and study them as a function of a single parameter that closely approximates the signal-to-noise ratio (SNR). Using this approach, we show that virtually all the iterative decoding schemes in use today exhibit similar qualitative dynamics. In particular, a whole range of phenomena known to occur in nonlinear systems, such as existence of multiple fixed points, oscillatory behavior, bifurcations, chaos, and transient chaos are found in iterative decoding algorithms. As an application, we develop an adaptive technique to control transient chaos in the turbo-decoding algorithm, leading to a substantial improvement in performance. We also propose a new stopping criterion for turbo codes that achieves the same performance with considerably fewer iterations. Index Terms--Bifurcation theory, chaos, dynamical systems, iterative decoding, low-density parity-check codes, product codes, turbo codes.
    Keywords: Chaos Theory -- Analysis ; Bifurcation Theory -- Analysis ; Entropy (Information theory) -- Analysis
    ISSN: 0018-9448
    E-ISSN: 15579654
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  • 6
    In: IEEE Transactions on Circuits and Systems-I: Fundamental Theory..., March, 2001, Vol.48(3), p.382
    Description: This paper and its companion (Part I) are devoted to to the analysis of the application of a chaotic piecewise-linear one-dimensional (PL1D) map as Random Number Generator (RNG). In Part I, we have mathematically analyzed the information generation process of a class of PL1D maps. In this paper, we find optimum parameters that give an RNG with lowest redundancy and maximum margin against parasitic attractors. Further, the map is implemented in a 0.8 [micro]]m standard CMOS process utilizing switched current techniques. Post-layout circuit simulations of the RNG indicate no periodic attractors over variations in temperature, power supply and process conditions, and maximum redundancy of 0.4%. We estimate that the output bit rate of our RNG is 1 Mbit/s, which is substantially higher than the output bit rate of RNGs available on the market. Index Terms--Chaos, CMOS, random number generator.
    Keywords: Random Number Generators (Software) -- Models ; Complementary Metal Oxide Semiconductors -- Usage ; Chaotic Systems -- Usage
    ISSN: 1057-7122
    E-ISSN: 15581268
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  • 7
    Language: English
    In: IEEE Transactions on Circuits and Systems-I-Regular Papers, June, 2006, Vol.53(6), p.1341(12)
    Description: Digital chaotic ciphers have been investigated for more than a decade. However, their overall performance in terms of the tradeoff between security and speed, as well as the connection between chaos and cryptography, has not been sufficiently addressed. We propose a chaotic Feistel cipher and a chaotic uniform cipher. Our plan is to examine crypto components from both dynamical-system and cryptographicai points of view, thus to explore connection between these two fields. In the due course, we also apply dynamical system theory to create cryptographically secure transformations and evaluate cryptographical security measures. Index Terms--Block cipher, chaos, cryptography, differential probability, linear probability.
    Keywords: Cryptography -- Analysis ; Chaos Theory -- Analysis ; Ciphers -- Analysis
    ISSN: 1549-8328
    E-ISSN: 15580806
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  • 8
    In: IEEE Transactions on Circuits and Systems-I: Fundamental Theory..., Oct, 1995, Vol.42(10), p.779(6)
    Description: This work deals with Turing instability in two diffusely coupled systems. We show that for the certain type of vector fields, diffusive coupling can cause instability of otherwise stable fixed points, and produce chaos. Accompanying numerical examples are obtained using Chua's equation. We discuss possible application of this concept of instability in modeling of biological cell assemblies, as well as some generalizations that would be interesting to attempt.〈〈ETX〉〉
    Keywords: Nonlinear Theories -- Research ; Diffusion Processes -- Research ; Spatial Systems -- Research
    ISSN: 1057-7122
    E-ISSN: 15581268
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  • 9
    Language: English
    In: International Journal of Bifurcation and Chaos, August 2008, Vol.18(8), pp.2415-2423
    Description: Synchronization of chaotic systems with unknown parameters, or namely auto-synchronization, has aroused a lot of interest in the past few years. In this paper, we are interested in solving one of the critical cases in Lorenz system. Based on the past researches, it was concluded that the auto-synchronization of Lorenz system was inachievable if the unknown parameters resided in all of its three dynamical equations while only the first state was measurable. However, as demonstrated in this paper, this problem can be resolved by an adaptive observer following simple design rules. The design is justified with the conditional Lyapunov exponents, and its effectiveness is clearly illustrated in simulations.
    Keywords: Adaptive Observer ; Auto-Synchronization ; Lorenz System ; Engineering ; Sciences (General)
    ISSN: 0218-1274
    E-ISSN: 1793-6551
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  • 10
    Language: English
    In: IEEE Transactions on Circuits and Systems-I-Regular Papers, July, 2005, Vol.52(7), p.1382(12)
    Description: Chebyshev polynomials have been recently proposed for designing public-key systems. Indeed, they enjoy some nice chaotic properties, which seem to be suitable for use in Cryptography. Moreover, they satisfy a semi-group property, which makes possible implementing a trapdoor mechanism. In this paper, we study a public-key cryptosystem based on such polynomials, which provides both encryption and digital signature. The cryptosystem works on real numbers and is quite efficient. Unfortunately, from our analysis, it comes up that it is not secure. We describe an attack which permits to recover the corresponding plaintext from a given ciphertext. The same attack can be applied to produce forgeries if the cryptosystem is used for signing messages. Then, we point out that also other primitives, a Diffie-Hellman like key agreement scheme and an authentication scheme, designed along the same fines of the cryptosystem, are not secure due to the aforementioned attack. We close the paper by discussing the issues and the possibilities of constructing public-key cryptosystems on real numbers. Index Terms--Chaos theory, Chebyshev polynomials, Jacobian elliptic Chebyshev rational map, public-key cryptography.
    Keywords: Public Key Encryption -- Research ; Chaos Theory -- Research
    ISSN: 1549-8328
    E-ISSN: 15580806
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