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  • Szczepanski, Janusz  (7)
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  • 1
    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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  • 2
    Language: English
    In: Physical review letters, 03 December 2004, Vol.93(23), pp.234101
    Description: We propose a definition of finite-space Lyapunov exponent. For discrete-time dynamical systems, it measures the local (between neighboring points) average spreading of the system. We justify our definition by showing that, for large classes of chaotic maps, the corresponding finite-space Lyapunov exponent approaches the Lyapunov exponent of a chaotic map when M--〉infinity, where M is the cardinality of the discrete phase space. In analogy with continuous systems, we say the system has pseudochaos if its finite-space Lyapunov exponent tends to a positive number (or to +infinity), when M--〉infinity.
    Keywords: Physics;
    ISSN: 0031-9007
    E-ISSN: 10797114
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  • 3
    Language: English
    In: IEEE Transactions on Circuits and Systems-II-Express Briefs, Oct, 2007, Vol.54(10), p.882(5)
    Description: In a recent paper, Jakimoski and Subbalakshmi provided a nice connection between the so-called discrete Lyapunov exponent of a permutation F defined on a finite lattice and its maximal differential probability, a parameter that measures the complexity of a differential cryptanalysis attack on the substitution defined by F. In this brief, we take a second look at their result to find some practical shortcomings. We also discuss more general aspects. Index Terms--Differential cryptanalysis, discrete Lyapunov exponent (DLE), maximum differential probability (DP).
    Keywords: Liapunov Functions -- Evaluation ; Probability Theory -- Research ; Cryptography -- Research
    ISSN: 1549-7747
    E-ISSN: 15583791
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  • 4
    Language: English
    In: Physics Letters A, 2006, Vol.355(1), pp.27-31
    Description: Chaotic maps can mimic random behavior in a quite impressive way. In particular, those possessing a generating partition can produce any symbolic sequence by properly choosing the initial state. We study in this Letter the ability of chaotic maps to generate order patterns and come to the conclusion that their performance in this respect falls short of expectations. This result reveals some basic limitation of a deterministic dynamic as compared to a random one. This being the case, we propose a non-statistical test based on ‘forbidden’ order patterns to discriminate chaotic from truly random time series with, in principle, arbitrarily high probability. Some relations with discrete chaos and chaotic cryptography are also discussed.
    Keywords: Chaotic Maps ; Order Patterns ; Permutation Entropy ; Discrete Lyapunov Exponent ; Chaotic Cryptography ; Physics
    ISSN: 0375-9601
    E-ISSN: 1873-2429
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  • 5
    Language: English
    In: IEEE Transactions on Circuits and Systems-I-Regular Papers, Feb, 2005, Vol.52(2), p.443(11)
    Description: In this paper, we explore, following Shannon's suggestion that diffusion should be one of the ingredients of resistant block ciphers, the feasibility of designing cryptographically secure substitutions (think of S-boxes, say) via approximation of mixing maps by periodic transformations. The expectation behind this approach is, of course, that the nice diffusion properties of such maps will be inherited by their approximations, at least if the convergence rate is appropriate and the associated partitions are sufficiently fine. Our results show that this is indeed the case and that, in principle, block ciphers with close-to-optimal immunity to linear and differential cryptanalysis (as measured by the linear and differential approximation probabilities) can be designed along these guidelines. We provide also practical examples and numerical evidence for this approximation philosophy. Index Terms--Black cipher, differential cryptanalysis, linear cryptanalysis, mixing dynamical system, periodic approximation, S box.
    Keywords: Ciphers -- Analysis ; Cryptographic Equipment -- Design And Construction
    ISSN: 1549-8328
    ISSN: 10577122
    E-ISSN: 15580806
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  • 6
    Language: English
    In: Physics Letters A, 2005, Vol.343(1), pp.55-60
    Description: We show that chaotic maps may be used for designing so-called substitution boxes for ciphers resistant to linear and differential cryptanalysis, providing an alternative to the algebraic methods. Our approach is based on the approximation of mixing maps by periodic transformations. The expectation behind is, of course, that the nice chaotic properties of such maps will be inherited by their approximations, at least if the convergence rate is appropriate and the associated partitions are sufficiently fine. We show that this is indeed the case and that, in principle, substitutions with close-to-optimal immunity to linear and differential cryptanalysis can be designed along these guidelines. We provide also practical examples and numerical evidence for this approximation philosophy.
    Keywords: Chaotic Maps ; Periodic Approximations ; Bit Permutations ; Cryptanalysis ; Physics
    ISSN: 0375-9601
    E-ISSN: 1873-2429
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  • 7
    Language: English
    In: Physics Letters A, 2008, Vol.372(41), pp.6265-6268
    Description: One of the possible by-products of discrete chaos is the application of its tools, in particular of the discrete Lyapunov exponent, to cryptography. In this Letter we explore this question in a very general setting.
    Keywords: Physics
    ISSN: 0375-9601
    E-ISSN: 1873-2429
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