*4.2. Chaos-Based Pseudorandom Number Generator*

A chaotic map could be a suitable source to generate pseudorandom numbers when it has high sensitivity, good ergodicity, and extreme unpredictability. The existence of these features in a chaotic map can be determined by the NIST-800-22 randomness tests.

It is, therefore, crucial to determine the existence of such features in the 2D-ICSM to examine its ability to be a PRNG. In this regard, we propose a simple strategy, which directly employs the chaotic sequences as pseudorandom numbers by converting each of their values to a 32-bit binary stream using the IEEE 754 float standard. Figure 6 displays the NIST SP800-22 test results of pseudorandom numbers generated by the 2D-ICSM. In this figure, the generated sequence by 2D-ICSM has a length of 100, 000, 000 binary bits. It is important to state here that a chaotic map can pass the statistical tests of NIST-800-22 only when the corresponding p-values are greater than the experimental significance level [40]. Consequently, the results in Figure 6 demonstrates the high randomness of the generated pseudo random numbers by the 2D-ICSM.

**Figure 6.** The p-values of the binary sequence generated by PRNG of the 2D-ICSM with the parameters *α* = 6 and *β* = 12.

#### **5. Complexity-Based Sample Entropy**

In this section, the complexity of 2D-ICSM is investigated through a fundamental algorithm, namely, Sample Entropy (SamEn). The authors of [36] presented SamEn to calculate how much extra information is required to predict the (*t* + 1)th output of a trajectory using its previous (*t*) outputs. SamEn with larger values indicate a lower degree of regularity of a chaotic map. In other words, the chaotic map exhibits a high level of complexity and unpredictability.

The SamEn algorithm for a given time series {*x*(*i*), *i* = 0, 1, 2, . . . , *N* − 1} is outlined as follows:

1. Reconstruction: the time series can be reconstructed as follows,

$$X\_i = \{ \mathbf{x}\_{i\prime} \mathbf{x}\_{i+\tau\prime} \dots \mathbf{x}\_{i+(m-1)\tau} \}, \quad X\_i \in R^m \tag{5}$$

where *m* is embedding dimension and *τ* is time delay.

2. Counting the vector pairs: For a given tolerance parameter *r*, let *Bi* be the number of vectors *Xj* such that

$$d[X\_i, X\_j] \le r, \quad i \ne j \tag{6}$$

here, *d*[*Xi*, *Xj*] is the distance between *Xi* and *Xj*, which is defined as

$$d[X\_i, X\_j] = \max\{|\mathbf{x}(i+k) - \mathbf{x}(j+k)|: \\ \begin{aligned} \mathbf{x}(j+k)|: \\ 0 \le k \le m-1 \end{aligned} \}\tag{7}$$

3. Calculating *θm*(*r*): According to the obtained number of vector pairs, we can get

$$C\_i^m(r) \quad = \frac{B\_i}{N - (m-1)\tau'} \tag{8}$$

then calculate *θm*(*r*) by

$$\theta^m(r) \quad = \frac{\frac{\sum\_{i=1}^{N-(m-1)\tau} \ln C\_i^m(r)}{[N-(m-1)\tau]}}{[N-(m-1)\tau]}.\tag{9}$$

4. Calculating SamEn: Repeating the above steps we can get *θm*+1(*r*), then SamEn is given by

$$
\delta S m \to (m, r, N) = \theta^m(r) - \theta^{m+1}(r). \tag{10}
$$

Figure 7a plots SamEn results of the 2D-ICSM when the two parameters *α* and *β* are varying simultaneously. This figure provides a more clear vision of the complexity of 2D-ICSM. It can be seen from this figure that the 2D-ICSM exhibits high complexity in most of its parameters setting. However, the highest SamEn values appear whenever the *α* and *β* are increasing.

Moreover, Figure 7b depicts the SamEn results of the 2D-ICSM and different chaotic and hyperchaotic maps. It is quite clear that the 2D-ICSM has the largest SamEn values, which indicates that one needs more information to predict the generated sequences by this map.

**Figure 7.** SamEn simulation: (**a**) SamEn values of the 2D-ICSM when its parameters vary; (**b**) SamEn results of different chaotic and hyperchaotic maps, where parameter *φ* represents *α*, *a*, *a*2, *α*1, for the 2D-ICSM, 2D-LASM [14], 2D-SIMM [13], and 2D-SLMM [12], respectively.

#### **6. Chaos Based Cryptography**

This section investigates the performance of 2D-ICSM in cryptography applications by designing a symmetric secure communication system. Figure 8 displays the schematic diagram of the proposed symmetric secure communication scheme. As can be observed in this figure, the proposed communication system is designed to transmit a message *M*(*s*) between two points in which the 2D-ICSM is employed to encrypt the information.

**Figure 8.** Schematic diagram of the proposed secure communication scheme using the 2D-ICSM.
