On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

When chaotic systems are realized with finite precisions in digital computers, their dynamical properties are often found to be entirely different from the original versions in the continuous setting. In the literature, there does not seem to be much work on quantitative analysis of such degradation of digitized chaos and how to reduce its negative influence on chaos-based digital systems. Focusing on 1D piecewise linear chaotic maps (PWLCM), this paper reports some findings on a new series of dynamical indicators, which can quantitatively reflect the degradation effects on a digital PWLCM realized with a fixed-point finite precision. On top of that, the paper introduces a new method for studying digital chaos from an algorithmic point of view. In addition, the theoretical results obtained in this paper should be very helpful for the consideration of reducing negative influence of dynamical degradation in real design of various digital chaotic systems. As typical examples, the proposed dynamical indicators are applied to the performance comparison of different remedies for improving dynamical degradation, cryptanalysis of digital chaotic ciphers based on 1D PWLCM, and design of chaotic pseudo-random number generators with desired characteristics.

[1]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[2]  N. Meyers,et al.  H = W. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[3]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[4]  Larry L. Dornhoff,et al.  Applied Modern Algebra , 1978 .

[5]  G. Benettin,et al.  On the reliability of numerical studies of stochasticity , 1979 .

[6]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[7]  Y. Lévy.  Some remarks about computer studies of dynamical systems , 1982 .

[8]  Charles F. F. Karney.  Long-time correlations in the stochastic regime , 1983, nlin/0501023.

[9]  Hogg,et al.  Attractors on finite sets: The dissipative dynamics of computing structures. , 1985, Physical review. A, General physics.

[10]  W. F. Wolff,et al.  Transients and asymptotics in granular phase space , 1986 .

[11]  Binder,et al.  Simulating chaotic behavior with finite-state machines. , 1986, Physical review. A, General physics.

[12]  J. Palmore,et al.  Computable chaotic orbits , 1986 .

[13]  François Robert,et al.  Discrete iterations - a metric study , 1986, Springer series in computational mathematics.

[14]  C. Beck,et al.  Effects of phase space discretization on the long-time behavior of dynamical systems , 1987 .

[15]  J. Palmore,et al.  Shadowing by computable chaotic orbits , 1987 .

[16]  I. Percival,et al.  Arithmetical properties of strongly chaotic motions , 1987 .

[17]  K. Kaneko.  Symplectic cellular automata , 1988 .

[18]  Grebogi,et al.  Roundoff-induced periodicity and the correlation dimension of chaotic attractors. , 1988, Physical review. A, General physics.

[19]  Leon O. Chua,et al.  Chaos in digital filters , 1988 .

[20]  A. Boyarsky,et al.  Why computers like lebesgue measure , 1988 .

[21]  E. Thiran,et al.  p-adic dynamics , 1989 .

[22]  Daniel D. Wheeler,et al.  Problems with Chaotic Cryptosystems , 1989, Cryptologia.

[23]  Robert A. J. Matthews,et al.  On the Derivation of a "Chaotic" Encryption Algorithm , 1989, Cryptologia.

[24]  J. Palmore,et al.  Computer arithmetic, chaos and fractals , 1990 .

[25]  I. Sasase,et al.  A Secret Key Cryptosystem Using a Chaotic Map , 1990 .

[26]  J. Keating.  Asymptotic properties of the periodic orbits of the cat maps , 1991 .

[27]  Daniel D. Wheeler,et al.  Supercomputer Investigations of a Chaotic Encryption Algorithm , 1991, Cryptologia.

[28]  L. Chua,et al.  On chaos of digital filters in the real world , 1991 .

[29]  Iwao Sasase,et al.  A Secret Key Cryptosystem by Iterating a Chaotic Map , 1991, EUROCRYPT.

[30]  Slawomir T. Fryska,et al.  Computer dynamics and shadowing of chaotic orbits , 1992 .

[31]  P. Binder.  Limit cycles in a quadratic discrete iteration , 1992 .

[32]  D. Earn,et al.  Exact numerical studies of Hamiltonian maps: iterating without roundoff error , 1992 .

[33]  Sean Murphy,et al.  Henon stream cipher , 1992 .

[34]  Michael Ganley,et al.  Encryption algorithms , 1992 .

[35]  Fengi Hwu,et al.  The interpolating random spline cryptosystem and the chaotic-map public-key cryptosystem , 1993 .

[36]  P. Borcherds,et al.  The digital tent map and the trapezoidal map , 1993 .

[37]  L. Chua,et al.  On chaos in digital filters: Case b = -1 , 1993 .

[38]  M. Blank.  Pathologies generated by round-off in dynamical systems , 1994 .

[39]  Bernard P. Zajac.  Applied cryptography: Protocols, algorithms, and source code in C , 1994 .

[40]  D. Arrowsmith,et al.  Geometry of p -adic Siegel discs , 1994 .

[41]  Franco Vivaldi,et al.  Periodicity and Transport from Round-Off Errors , 1994, Exp. Math..

[42]  P. Kloeden,et al.  An invariant measure arising in computer simulation of a chaotic dynamical system , 1994 .

[43]  Clare D. McGillem,et al.  A chaotic direct-sequence spread-spectrum communication system , 1994, IEEE Trans. Commun..

[44]  P. Kloeden,et al.  Collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre , 1995 .

[45]  A. Baranovsky,et al.  DESIGN OF ONE-DIMENSIONAL CHAOTIC MAPS WITH PRESCRIBED STATISTICAL PROPERTIES , 1995 .

[46]  C. Wu,et al.  Complex behavior in digital filters with overflow nonlinearity: analytical results , 1996 .

[47]  H. Waelbroeck,et al.  Discrete Chaos , 1996, chao-dyn/9610005.

[48]  J. Cernák.  Digital generators of chaos , 1996 .

[49]  L. Chua,et al.  CLARIFYING CHAOS: EXAMPLES AND COUNTEREXAMPLES , 1996 .

[50]  M. Blank.  Discreteness and Continuity in Problems of Chaotic Dynamics , 1997 .

[51]  Hong Zhou,et al.  Problems with the chaotic inverse system encryption approach , 1997 .

[52]  Hong Zhou,et al.  Secure communication via one-dimensional chaotic inverse systems , 1997, Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97.

[53]  Z. Hong,et al.  Generating Chaotic Secure Sequences with Desired Statistical Properties and High Security , 1997 .

[54]  F. Vivaldi,et al.  ANOMALOUS TRANSPORT IN A MODEL OF HAMILTONIAN ROUND-OFF , 1998 .

[55]  J. Fridrich.  Symmetric Ciphers Based on Two-Dimensional Chaotic Maps , 1998 .

[56]  Sang Tao,et al.  Perturbance-based algorithm to expand cycle length of chaotic key stream , 1998 .

[57]  Tao Sang,et al.  Clock-controlled chaotic keystream generators , 1998 .

[58]  F. Vivaldi,et al.  Small perturbations of a discrete twist map , 1998 .

[59]  M. Mackey,et al.  Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics , 1998 .

[60]  E. Alvarez,et al.  New approach to chaotic encryption , 1999 .

[61]  F. Vivaldi,et al.  An algorithmic view of pseudochaos , 1999 .

[62]  Kiyoshi Tanaka,et al.  Truncated Baker transformation and its extension to image encryption , 1999, Optics + Photonics.

[63]  W. G. Chambers.  ORBIT-PERIODS IN SECOND-ORDER FINITE-PRECISION DIGITAL FILTERS WITH OVERFLOW , 1999 .

[64]  Li-Hai Zhou,et al.  A new idea of using one-dimensional PWL map in digital secure communications-dual-resolution approach , 2000 .

[65]  A. Pokrovskii,et al.  Mixed moments of random mappings and chaotic dynamical systems , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[66]  E. Weisstein.  Total Probability Theorem , 2000 .

[67]  D. Bosio,et al.  Round-off errors and p-adic numbers , 2000 .

[68]  Mieczyslaw Jessa,et al.  Data encryption algorithms using one-dimensional chaotic maps , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[69]  L. Kocarev,et al.  Logistic map as a block encryption algorithm , 2001 .

[70]  Xuanqin Mou,et al.  Pseudo-random Bit Generator Based on Couple Chaotic Systems and Its Applications in Stream-Cipher Cryptography , 2001, INDOCRYPT.

[71]  Shujun Li,et al.  Statistical Properties of Digital Piecewise Linear Chaotic Maps and Their Roles in Cryptography and Pseudo-Random Coding , 2001, IMACC.

[72]  Anastasios Bezerianos,et al.  A Probabilistic Symmetric Encryption Scheme for Very Fast Secure Communication Based on Chaotic Systems of difference equations , 2001, Int. J. Bifurc. Chaos.

[73]  L. Kocarev,et al.  Chaos and cryptography: block encryption ciphers based on chaotic maps , 2001 .

[74]  X. Mou,et al.  Improving security of a chaotic encryption approach , 2001, Physics Letters A.

[75]  Kazuyuki Aihara,et al.  Cryptosystems based on space-discretization of chaotic maps , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[76]  W. Schwarz,et al.  Chaos and cryptography , 2001 .

[77]  Ninan Sajeeth Philip,et al.  Chaos for Stream Cipher , 2001, ArXiv.

[78]  Mieczyslaw Jessa,et al.  Data transmission with adjustable security exploiting chaos-based pseudorandom number generators , 2002, 2002 IEEE International Symposium on Circuits and Systems. Proceedings (Cat. No.02CH37353).

[79]  K. Yano,et al.  Image Encryption Scheme Based on a Truncated Baker Tranformation , 2002 .

[80]  K. Aihara,et al.  Cryptosystems with discretized chaotic maps , 2002 .

[81]  Chee Kheong Siew,et al.  A new block cipher based on chaotic tent maps , 2002 .

[82]  P. García,et al.  Communication through chaotic map systems , 2002 .

[83]  Kazuyuki Aihara,et al.  Dynamical Characteristics of Discretized Chaotic Permutations , 2002, Int. J. Bifurc. Chaos.

[84]  K. Yano,et al.  Image Encryption Scheme Based on a Truncated Baker Transformation , 2002, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[85]  Xuanqin Mou,et al.  Chaotic encryption scheme for real-time digital video , 2002, IS&T/SPIE Electronic Imaging.

[86]  Z. Ji,et al.  Cryptanalysis of a Class of Chaotic Stream Ciphers (一类混沌流密码的分析) , 2003 .

[87]  X. Mou,et al.  On the security of a chaotic encryption scheme: problems with computerized chaos in finite computing precision , 2003 .

[88]  Shujun Li,et al.  Analyses and New Designs of Digital Chaotic Ciphers , 2003 .

[89]  X. Mou,et al.  Performance analysis of Jakimoski–Kocarev attack on a class of chaotic cryptosystems , 2003 .

[90]  GUANRONG CHEN,et al.  Can a Three-Dimensional Smooth Autonomous Quadratic Chaotic System Generate a Single Four-scroll Attractor? , 2004, Int. J. Bifurc. Chaos.

[91]  Guanrong Chen,et al.  Cryptanalysis of a Chaotic Neural Network Based Multimedia Encryption Scheme , 2004, PCM.

[92]  Guanrong Chen,et al.  Cryptanalysis of RCES/RSES Image Encryption Scheme , 2004, IACR Cryptol. ePrint Arch..

[93]  Shujun Li,et al.  When Chaos Meets Computers , 2004, nlin/0405038.