Statistical Properties of Digital Piecewise Linear Chaotic Maps and Their Roles in Cryptography and Pseudo-Random Coding

Abstract

Abstract. The applications of digital chaotic maps in discrete-time chaotic cryptography and pseudo-random coding are widely studied recently. However, the statistical properties of digital chaotic maps are rather different from the continuous ones, which impedes the theoretical analyses of the digital chaotic ciphers and pseudo-random coding. This paper detailedly investigates the statistical properties of a class of digital piecewise linear chaotic map (PLCM), and rigorously proves some useful results. Based on the proved results, we further discuss some notable problems in chaotic cryptography and pseudo-random coding employing digital PLCM-s. Since the analytic methods proposed in this paper can essentially extended to a large number of PLCM-s, they will be valuable for the research on the performance of such maps in chaotic cryptography and pseudo-random coding.

On the Dynamical Degradation of Digital Piecewise Linear Chaotic Maps

“…Actually, motivated by various "strange" phenomena of chaos observed on digital computers and in numerical simulations, pathologies of digital chaotic systems have been observed and extensively studied in the field of chaos theory [Arrowsmith & Vivaldi, 1994;Beck & Roepstorff, 1987;Benettin et al, 1978;Binder, 1992;Binder & Jensen, 1986;Blank, 1994Blank, , 1997Borcherds & McCauley, 1993;Bosioand & Vivaldi, 2000;Chambers, 1999;Chirkikov & Vivaldi, 1999;Diamond et al, 1994Diamond et al, , 1995Earn & Tremaine, 1992;Fryska & Zohdy, 1992;Góra & Boyarsku, 1988;Grebogi et al, 1988;Hogg & Huberman, 1985;Huberman, 1986;Kaneko, 1988;Karney, 1983;Keating, 1991;Levy, 1982;Li et al, 2001a;Lowenstein & Vivaldi, 1998;Masuda & Aihara, 2002b;McCauley & Palmore, 1986;Palmore & Herring, 1990;Palmore & McCauley, 1987;Percival & Vivaldi, 1987;Pokrovskii et al, 1999;Rannou, 1974;Thiran et al, 1989;Čermák, 1996;Vivaldi, 1994;Waelbroeck & Zertuche, 1999;Zhang & Vivaldi, 1998]. To show how such dynamical degradation occurs, assume that the discretized space has ...…”

“…As the main goal of this paper, a general framework will be introduced for studying digital chaos generated by piecewise linear chaotic maps (PWLCM) from an algorithmic point of view, which is an extension of our early work reported in [Li et al, 2001a]. For digital PWLCM, a new series of dynamical indicators are found to quantitatively measure their dynamical degradation under (finite-precision) fixed-point arithmetic.…”

Modified Logistic Maps for Cryptographic Application

“…There is superior sensitivity to initial condition. According to Figure 11 and Figure 12, SML map is chaotic when parameter "r" lies in intervals [2,4].…”

“…Lyapunov exponent of SML map with respect to parameter 'r' are calculated and plotted in Figure 12. According to Figure 11 and Figure 12, SML map is chaotic when parameter 'r' lies in intervals [2,4]. …”

“…It is used in chaos-based secure communication system and for generations of binary numbers. Li, Mou, and Cai proposed statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding [4]. Addabbo proposed performance analysis and optimized design of piecewise linear chaotic maps such as digital saw-tooth and tent maps as a source of pseudo-random bits generators [5] [6].…”