## 1 Introduction
In this extensive variety of proffers, several implementations were identified with security disadvantages attributed to one or more of the following features: non-uniform statistical distribution [49], digital degradation [50] and predictability [51] of the produced number sequences, as well as the small-sized seed space of the chaotic map [21]. In this context, other authors have proposed alternative solutions to counteract the exposed disadvantages. For example, in 2016, Wang et al. [52] compared cryptographically useful properties of piecewise-linear maps (ergodicity, Lyapunov exponent and bifurcation) to properties of logistic map and, in order to overcome the disadvantages of the logistic map used in designs of chaos-basedciphers, they proposed a PRNG based on the piecewise-logistic map. In that proffer, Wang et al. claim that their PRNG achieves a trade-off between efficiency and security. However, in 2019, Lambic [53] analyzed the security of PRNG based on the piecewise-logistic map showing that it can be violated by using brute-force and known output sequence attacks. And then, also in 2019, Wang et al. [49] proposed a four-dimensional chaotic model based on a piecewise-logistic map with coupled parameters, but they just tried to overcome the fact that the statistical distribution of the piecewise-logistic map is non-uniform. Another example showing a solution to overcome the security disadvantages revealed for PRNGs based on chaotic maps is the work proposed by Zhou et al. [54] in 2016. Zhou et al. proposed a secret key generation algorithm based in operations in the YUV color space that combines two secret keys to produce the initial conditions required in the chaotic maps used in the encryption processes. In the attempt to strengthen their encryption system against differential attacks, Zhou et al. used a cubic map and a wavelet map to produce pseudorandom number sequences. Although Zhou et al. considered these maps to be highly sensitive to initial conditions, they did not perform a formal sensitivity analysis. Also, in 2021, Shi and Deng [55] when studying the dynamical degradation of the two-dimensional Barker map they found that this chaotic map can have valuable properties when it is used in a PRNG. Another example showing the application of strategies to overcome security disadvantages in PRNGs based on chaotic maps is the work proposed by Murillo-Escobar et al. in 2017 [36]. Since under the premise that low-dimensional chaotic systems may become more used than high-dimensional chaotic systems to produce the pseudorandom key stream used for encryption purposes, Murillo-Escobar et al. [36] proposed a PRNG based on the pseudorandomly enhanced logistic map, claiming that the produced number sequences have excellent statistical properties to cryptography applications. Although Murillo-Escobar et al. specified that the parameter domain for pseudorandomly enhanced logistic map is limited to (3.999, 4.0), they scaled and discretized the output of the chaotic map by applying _mod 1_ to it when \(1\times 10^{6}\) is the scaling factor. With this scaling factor, they intended a uniform statistical distribution of the generated number sequences. Also, although Murillo-Escobar et al. [36] claim to avoid weak keys in their PRNG, we emphasize that they did not identify which conditions cause the chaotic map to produce weak keys in order to avoid them. A last example related to overcome the security disadvantages of the PRNGs based on chaotic maps is the work proposed by Chen et al. in 2019 [50]. Chen et al. [50] proposed a method to counteract the dynamical degradation of the digital sequences produced by using a chaotic system when it is implemented on low-precision devices; in that condition, all the produced sequences could be periodic sequences. In this way, the method proposed by Chen et al. [50] was based on a dynamical strategy to perturb a digital chaotic system by using pseudorandom sequences produced by a two-dimensional sine chaotic map with control parameters \(a\) and \(b\).