## 1 Introduction
They specify \(a=1\) and \(b=5\) so that the map has a chaotic behavior, but they do not perform an analysis of the opportunities that exist to generate chaos, nor of the chaos annulling conditions in the chaotic system. Additionally, Chen et al. [50] showed two experiments in order to test effectiveness of their method to counteract the dynamical degradation of digital chaotic sequence. In the first experiment they selected the logistic map to represent the one-dimensional chaotic maps. In the second experiment, they selected the two-dimensional logistic cascade hyperchaotic map to represent the high-dimensional chaotic maps. In this way, Chen et al. [50] demonstrated the effectiveness of their method considering the linear complexity, correlation, and statistical distribution. Therefore, although efforts are being made to overcome the security disadvantages of implementing PRNGs based on chaotic maps, there are still PRNGs based on chaotic maps that have security shortcomings. For example, some chaotic maps have stability islands within the parameter domains for chaotic behavior, adversely affecting the system security, other chaotic maps produce number sequences with non-uniform statistical distribution, and other chaotic maps only work by using a limited size of initial conditions space [15; 53]. Therefore, to safely use a PRNG based on chaotic maps, we must carefully select the initial conditions ensuring that the map will always produce pseudorandom sequences with uniform statistical distribution and it will operate into the parameter domains for chaotic behavior avoiding the annulling chaos conditions; and when the chaotic system is implemented electronically, the dynamic degradation of the digital sequences must be considered. Focusing specifically on PRNGs based on a single chaotic map, the most commonly used systems to generate pseudorandom number sequences are one-dimensional (1-D) chaotic maps, and although they have security disadvantages when used in cryptography, they are commonly used due to their structural simplicity, discrete nature, reduced number of arithmetic operations, high performance processing, and relatively easy implementation in hardware and software. It is worth noting that the 1-D chaotic maps can be attacked using the non-linear prediction method based on phase space reconstruction. In fact, in 1994 Short [51] proposed a method that can attack almost all 1-D chaotic maps and, therefore, many authors of works related to chaos-based PRNGs tend to conclude that it is more appropriate to use high-dimensional (H-D) chaotic systems rather than low-dimensional (L-D) chaotic systems to build PRNGs. It should be also noted that Short indicated in [51] that the details of their nonlinear prediction method is in a work submitted to the _Int. J. Bifurcations and Chaos_ since 1993, but it was not published. Instead of that work, there is another work published in 1997 by Short [56] that applies the non-linear dynamic prediction to extract, in the time domain, faithful representations of hidden message signals transmitted by chaotic communication systems. Short's experiments are based on two fundamental facts. The first fact is that two systems (transmitter and receiver) implemented to reproduce the dynamic of a chaotic system can be synchronized without transmitting information related to their initial state. The second fact was that the ability of the receiver to synchronize with the transmitter is not affected by the addition of a low-powered message on the chaotic carrier. This means that, once synchronization is achieved, the chaotic carrier can be removed to reveal the message.