*4.1. RNG Algorithm Design and NIST 800-22 Test Results*

In this subsection, an RNG algorithm design is developed via the newly introduced chaos in order to obtain the random numbers to be employed in the algorithm. The design process of the RNG algorithm is carried out as exposed in Algorithm 1 (see Appendix A). In the design process of the RNG algorithm, firstly the initial states and parameters of the chaotic system are defined. Then, the sampling interval of the system is determined and the chaotic system is considered by using

fourth-order Runge-Kutta (RK-4) integration algorithm using this sampling value. As an outcome of the system analysis, float values are found for each cycle from each phase. On the float values obtained from each phase, the step values of the decimal parts after the comma are subjected to the mode 2 operation. As a result, 15 bits are generated from each phase in the each cycle. Further, these obtained values are added to the number sequences for each phase (rngx, rngy, rngz). This process continues for each number sequence until the 1 M. bit is generated for the NIST 800-22 randomness examinations. Because at least 1 M. bit is needed for NIST 800-22 tests. After 1 M bits are generated from each phase, the phases are subjected to Exclusive Or (XOR) processing in binary form and new random number sequences are generated as named rngxy, rngxz, rngyz, rngxyz in the Algorithm 1. The generated values from the x and y phases are subjected to XOR processing to obtain a rngxy random number sequence. Similarly, generated from the phases y and z for rngyz, the *x* and *z* phases for rngyz and the x, y, z phases for rngxyz are subjected to XOR processing. Finally, NIST 800-22 tests are employed to all obtained random bit sequences. When random bit sequences are tested singularly, they cannot pass some tests. For this reason, random bit sequences generated are subjected to 2 or 3 XOR operations.

For the safe use of random numbers, they must have an appropriate randomness. The NIST 800-22 tests are a set of internationally accepted and frequently used tests in the literature that define the numbers' randomness via a variety of different tests. The NIST 800-22 test outcomes for the random number sequences originated from the developed RNG algorithm are displayed in Table 1. According to the test outcomes, it is seen that all the random numbers created passed all the examinations.


**Table 1.** NIST 800-22 NIST Test Results.

#### *4.2. Voice Encryption Algorithm Design and Its Application*

A new voice encryption algorithm is developed via RNG algorithm introduced in the previous section, voice encryption usage and its analysis are performed. The block diagram of the encryption process is presented in Figure 12. In the encryption algorithm, after entering initial states and parameters of chaotic system, these values are transmitted to the receiving side as a key for generation of the random number sequences to be employed in the decryption process. In order to realize the bit-based encryption, the values are obtained from the voice file consisting of float values with appropriate sampling step and converted into binary form. With random bit sequences past all NIST 800-22 randomness tests obtained from the RNG, the voice file in the binary form is encrypted. XOR operation is used in encryption process. After the encryption operation, the encrypted binary bit array is changed to the float form to generate the encrypted voice file. After the encrypted voice data is sent to the receiver side in this way, the decryption process is performed by applying the reverse operations in the encryption. Thus, the original voice file is attained on the receiving side.

**Figure 12.** The block diagram of encryption process.

The voice files in encryption procedure are shown in Figure 13. The original, encrypted and decrypted voice file is demonstrated in Figure 13a–c, respectively. When comparing the original and encrypted voice file in Figure 13a,b; it is seen that a very different file is gotten than the original and the encryption process is successful. When Figure 13a,c are examined, it is observed that the decryption procedure is successful. Figure 14 shows the frequency spectrum analysis results of the encryption process. Frequency spectrum analysis is carried out to determine the frequency range of voice files. To determine the success of the encryption process, frequency spectrum analyzes are performed on original and encrypted voice files. The spectrum analysis results of original and encrypted voice files are illustrated in Figure 14a,b. If we compare these two graphs, it seems that the encrypted voice file has a rather wide frequency spectrum range than the original. When these results are evaluated, it shows the success of the encryption process.

**Figure 13.** The voice file (**a**) original; (**b**) encrypted; (**c**) decrypted.

**Figure 14.** The spectrum analysis outcomes of original and encrypted voice files. (**a**) Original, (**b**) Encrypted.

#### **5. Conclusions and Discussion**

In this article, a new chaotic system with hyperbolic sinusoidal nonlinearity is designed. The proposed system belongs to a new category of dynamical systems with hidden chaotic flows, which assist in further understanding of chaotic attractors and also to use them in interesting applications like cryptography and secure communication schemes. The feature of hidden chaotic attractors, such as in systems with line of equilibria or in systems with no equilibrium point, makes them more suitable for the aforementioned applications, due to the fact that using systems with hidden attractors adds complexity to the dynamical system, which is used in this kind of applications. Therefore, in this work a voice encryption scheme, which is based on the specific systems was studied. Based on the variations of parameters of the system, this flow presented two classes of hidden attractors (with line of equilibria and no equilibrium point) plus a self-excited attractor, which has been reported to the literature for the first time. Dynamical behavior of the proposed system was explored and its bifurcation diagram and spectrum of Lyapunov exponents were propounded. For the appropriate selection of the parameters, the flow could display periodic oscillations and double-scroll chaos attractors. The system's electronic simulation investigated the confirmation of the double-scroll chaos attractor in real word. Via the proposed chaotic system, a novel RNG design was realized and random number generation was performed. NIST 800-22 randomness examinations were employed to the produced numbers and it was determined that all tests passed. By using RNG design, a novel voice encryption algorithm was established and encryption process was done. Frequency spectrum analysis of the voice encryption procedure was executed. In line with the analysis outcomes, it has been found that the new RNG design produces high random numbers and that the suggested encryption algorithm effectively achieves the encryption process. Therefore, authors aim with this work to attract the interest of the research community in the use of chaotic dynamical systems with hidden attractors in encryption schemes, as the results are proved to be very promising. Finally, as a future plan, the hardware implementation of the specific approach has been planned.

**Author Contributions:** S.M. was responsible for formal analysis and writing—original draft preparation. C.K.V. and Ü.Ç. were responsible to conceptualization, methodology, and software. S.K. was responsible for software, supervision and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors of this paper declare that they have no conflicts of interest.

**Date Availability:** The MATLAB files data used to support the findings of this paper are available from the corresponding author upon request.
