## 1 Introduction  
Several works related to the PRNG design have been proposed; for example, there are strategies that implement PRNGs using linear feedback shift registers (LFSR) [1; 2; 3; 4; 5], while other strategies are based on block cipher [6], stream cipher [7], quantum walks [8], cellular automata [9; 10], chaotic oscillators and artificial neural networks (ANN) [11], or chaotic maps [12; 13; 14; 15]. There are also PRNG design approaches that combine several of the above strategies [16]. Considering this context, we focus our research on PRNGs based on chaotic maps. Chaotic maps are iterated functions that use an initial seed to produce non-linear sequences of numbers; these sequences when translated into binary sequences can generate random-looking and highly unpredictable numbers to be used in cryptography. Chaotic maps have high sensitivity to initial conditions when operating with their parameters inside specific domains, which can be determined. In these parameter domains the chaotic maps can operate as pseudorandom or aperiodic systems, but outside those parameter domains they can operate as periodic systems or their trajectories may also escape to infinity [17]. It may also happen that several chaotic repellers coexist in the chaotic system [18], and the trajectories move chaotically for a while before escaping and reaching another chaotic repeller [19; 20]. Under these considerations, the behavior option can be selected from their control parameters [14]. In addition, when the chaotic maps are used in cryptographic applications, several drawbacks become evident, such as range discontinuity and non-uniform statistical distribution of the generated number sequences, as well as the small seed space [21]. Despite this, we cannot forget that there is a natural application relationship between chaos and cryptography. The main features of the chaotic systems, such as the sensitive dependence on initial conditions and control parameters, ergodicity, size of the parameter space, and mixing property, can be related to the confusion and diffusion conditions that must be applied to information to be protected by using cryptographic systems [22]. Therefore, many cryptographic systems and modules have been proposed based on chaotic systems [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. In particular, chaotic maps have been successfully applied in the implementation of PRNGs [14; 15; 34; 35; 36; 37; 38; 39; 40; 41; 42]. In this way, the chaotic maps have inherent features that fully match with the practical implementation requirements of the PRNGs. The first proposal for a PRNG based on chaotic maps was developed in 1982 by Oishi and Inoue [43]. Later, Gonzalez and Pino in 1999 generalized the logistic map and designed a random function [44]. Stojanovski et al. in 2001 analyzed the application of a piecewise-linear chaotic map as PRNG [45; 46]; and in the same year, Li et al. [47] performed an analysis suggesting that a couple, \(g(f(x))\), of two piecewise-linear chaotic maps \(f(x)\) and \(g(x)\) has perfect cryptographic properties if it satisfies four requirements when used to build high-security stream ciphers. The requirements defined by Li et al. [47] for one-dimensional chaotic maps are: (R1 ) Piecewise-linear chaotic maps should be surjective maps on a same interval (a, b), (R2) Piecewise-linear chaotic maps should be ergodic on (a, b) with unique invariant density functions, (R3) Invariant density functions of the piecewise-linear chaotic maps should be equal to each other, and (R4) Chaotic orbit produced by one the piecewise-linear chaotic maps should be asymptotically independent to the chaotic orbit produced by the other map when the length of the chaotic orbits tends to be infinite. After the work developed by Li et al. [47], many other researchers proposed PRNGs based on chaotic maps using different approaches [12; 13; 14; 15; 34; 35; 36; 37; 38; 39; 40; 41; 42; 48].