## 2 Design of a Large-Scale CRNG  
In this paper, we will discuss the design and realization of a large-scale CRNGs and its application to secure communication based on synchronized master-slave large-scale CRNGs. To design the large-scale CRNGs, in the following, we first introduce the modulation of the Duffing map chaotic system such that diversified random chaotic state responses can be obtained for large-scale CRNGs. The original Duffing map system is described by the following system. \[ \begin{array}{l}x_{1}(k+1)=x_{2}(k)\\ x_{2}(k+1)=-0.2x_{1}(k)+2.75x_{2}(k)-x_{2}^{3}(k)\end{array}  \tag{1}\]  
where \(x_{1},x_{2}(k)\) are the system states. In order to adjust the amplitude and DC offset for state responses, we define new state variables \(y_{i},i=1,2\) satisfying  
\[y_{i}(k)=a_{i}x_{i}(k)+d_{i} \tag{2}\]  
where \(a_{i},i=1,2\) are given to adjust the amplitude and the DC offset of state responses, respectively. From Equation (2), we have  
\[x_{i}(k)=\frac{y_{i}(k)-d_{i}}{a_{i}},i=1,2 \tag{3}\]By (1) and (3), a new modulated Duffing map chaotic system can be obtained as  
\[ \begin{array}{l}y_{1}(k+1)=\lambda_{1}y_{2}(k)+\lambda_{2}\\ y_{2}(k+1)=\beta_{1}y_{1}(k)+\beta_{2}y_{2}^{3}+\beta_{3}y_{2}^{2}+\beta_{4}y_{ 2}+\beta_{5}\end{array}  \tag{4}\]  
where  
\[ \begin{array}{l}\lambda_{1}=\frac{a_{1}}{a_{2}},\lambda_{2}=-\frac{a_{1}}{a_ {2}}d_{2}+d_{1},\beta_{1}=-\frac{0.2a_{2}}{a_{1}}\\ \beta_{2}=-\frac{1}{a_{2}^{2}},\beta_{3}=\frac{3d_{2}}{a_{2}^{2}},\beta_{4}=2.75-\frac{3d_{2}^{2}}{a_{2}^{2}},\beta_{5}=\frac{0.2a_{2}d_{1}}{a_{1}}-2.75d_{ 2}+\frac{d_{2}^{3}}{a_{2}^{2}}+d_{2}\end{array} \]  
For evaluating the modulation of the system amplitude and DC offset of random number generator, we give the following simulation analysis. In the simulation, the modulation parameters are given as \(a_{1}=2,a_{2}=4,d_{1}=-5,d_{2}=5\) and the initial conditions are selected as \(x_{1}=0.4,x_{2}=-0.2\). The simulation results are given in Figures 1-3. Figure 1: The state responses of the Duffing map chaotic system (before modulation). Figure 2: The state responses of the Duffing map chaotic system (after modulation). According to the simulation results above, it reveals, as expected, the amplitude is modulated by 2 times and 4 times, while the DC offset is shifted by \(-5\) and \(5\). Furthermore, we use the National Institute of Standards and Technology (NIST) [18] test suite to test the randomness of the modulated chaotic states. In the NIST test, first, we set the test parameters including the sequences length \(n\) = 2 \(\times\) 10\({}^{7}\) bits, the number of subsequences, \(m\) = 20. Then, we take byte 5 (bits 32-39) of every modulated chaos states y1 and y2 in IEEE754 double-precision format [19] to test the randomness. Finally, all the test results are shown in Table 1. In Table 1, the outcome of the test values is called p value. When \(p\) value \(\geq\) 0.01, then it passes the test. From Table 1, the generated numbers of y1 pass all tests and those of y2 pass 13 tests. Therefore, we can conclude that the modulated chaotic states are with good randomness. From discussed above, the random number generator can be implemented by using the modulated Duffing map chaotic system. It is well known that the state response of a chaotic system is very sensitive to initial values and very small differences can lead to very different chaotic state responses, namely the butterfly effect. Based on this feature, we can use the butterfly effect to design large-scale CRNGs. In addition, due to the high capacity and fast computing speed of the digital microcontroller, we can implement this large-scale CRNG with microcontroller and can get a large amount of random numbers in the very short time according to the computing speed of the microcontroller. To show the butterfly effect, we use three sets of random generators (Duffing map systems in (1)) for comparison.