Properties that arise from a depth-zoom analysis of the Logistic map

Abstract can also be found here (página 2010)

Chaotic nonlinear dynamical systems have been widely applied in many domains of science, including Cryptography, which has obtained significant results mainly because of the sensitivity to initial conditions, ergodicity, and mixing properties that assure a high-quality pseudo-randomness, when looking for an efficient encryption that can be achieved throughout a proper construction [Arroyo]. Should be noticed that many researchers related to chaos theory employ computational implementations based on floating-point arithmetic, where every real number is an approximation. Thus, in a long-time, the expected trajectory may be away from the theoretical one, as a result of limited machine precision (discretization and truncation errors) [1]. We focused on one particular case of "failure" cryptographic algorithm based on the logistic map \(x^{t+1} = \mu x^{t}(1-x^{t})\) with interval (0,1), one of the most studied dynamical system and one of the firsts cryptosystems based on chaos. From a cryptographic viewpoint, this latter approach has been severely criticized [2,3], based on two points that may indicate insufficient pseudo-randomness: (i) Its probability distribution exhibits a "U" pattern, where is expected a plateau distribution as much as possible [2]; and (ii) Trajectories of short cycle length where is expected longer periodicity depending on machine limitations [3]. However, these arguments were based on a finite precision criteria (single- and double-precision) that may be underestimating the logistic map potential. This work focused in these two points of "failure" of the logistic map. This work explored each \(x^{t}\) number of a trajectory, in a depth-zoom manner, by using high-precision floating point arithmetic. This means, to discard the \(k\)-th digit to the left of the decimal separator in order to compound a specific \(x^{t}_{k}\) number. It is found that, in this proposed manner, an interesting phenomena appears. The generated sequences are not only more accurate, instead, it can be observed a rapid improvement of its probability distribution, achieving a more plateau distribution, and also an increase of the cycle lengths. Certainly, a profound analysis under this proposed manner is still needed, e.g., the bifurcation diagram, the phase diagram, the cobweb diagram, the Lyapunov exponents, cycles-length analysis, among others. From this first exploration can be observed some "folding and stretching" pattern of the \(x_{k}\)-th numbers on phase-space, which cannot be observed when traditional floating point arithmetic is used. Notwithstanding more analysis are still needed, these results may represent an interesting impact to chaos theory and cryptography, since this approach could be generalized to other discrete dynamical systems and consequently it may represent a chaotic-based encryption algorithm improvement. 

[1]       R. M. Corless, "What Good Are Numerical Simulations of Chaotic Dynamical Systems?," Comput. Math. Applic,, vol. 28, no. 19, pp. 107–121, 1994.

[2]       G. Álvarez, F. Montoya, M. Romera, and G. Pastor, "Cryptanalysis of an ergodic chaotic cipher," Phys. Lett. A, vol. 311, no. 2–3, pp. 172–179, 2003.

[3]       K. J. Persohn and R. J. Povinelli, "Analyzing logistic map pseudorandom number generators for periodicity induced by finite precision floating-point representation," Chaos, Solitons & Fractals, vol. 45, no. 3, pp. 238–245, 2012.