## 5 Conclusions  
In this work, a 3D chaotic system with a line equilibrium was proposed. The system was studied through the computation of its bifurcation diagrams and Lyapunov exponents diagram. To study the applicability of the system to encryption and security related applications, the system was applied to a secure communications scheme using the SCSK modulation method, yielding satisfactory BER performance. Moreover, the problem of random bit generation was considered. The proposed design can generate three bits per iteration, with statistical randomness for the generated sequence.  
It is worth noting here that as far as the numerical evaluation of the chaotic system in the transmitter and receiver ends is concerned, there are instances where errors can arise, that may lead to degradation of the design. Examples can include round-off errors that arise when a different error  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Statistical Test** & \(p\)**-Value** & **Proportion** & **Result** \\ \hline
8 & Non-Overlapping Template & 0.319084 & 50/50 & Success \\ \hline
9 & Overlapping Template & 0.137282 & 48/50 & Success \\ \hline
10 & Universal & 0.191687 & 50/50 & Success \\ \hline
11 & Approximate Entropy & 0.085587 & 50/50 & Success \\ \hline
12 & Random Excursions & 0.010606 & 29/29 & Success \\ \hline
13 & Random Excursions Variant & 0.186566 & 29/29 & Success \\ \hline
14 & Serial & 0.574903 & 50/50 & Success \\ \hline
15 & Linear Complexity & 0.262249 & 50/50 & Success \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 15: Autocorrelation and cross-correlation for a bit sequence of length \(10^{4}\).  
tolerance is chosen between transmitter and receiver, or errors that arise when a different integration step is chosen or different numerical methods are used to solve the system; see for example, discussions in the recent works [52; 53; 54]. This is one of the main obstacles that have so far held up the possible commercialization of chaos-based secure communications.  
Here, since no implementation of the design is performed, we assumed, as many works do, that the receiver and transmitter systems are simulated in devices with the same configuration settings. Yet, it is within our scope of research interest to physically implement such a design, to see when and how such erros arise and how they can be tackled. Thus, future aspects of this work would be the experimental implementation of the communications design [55], and the further application of the proposed PRBG to image encryption.  
Conceptualization, C.V., S.C., V.-T.P., L.M.; Formal analysis, L.M., C.V., S.C., V.-T.P., V.K.M.; Software, L.M., C.V., S.C.; Supervision, C.V., I.S., S.G., S.C., V.-T.P.; Validation, C.V., S.C., I.S., S.G., V.K.M.; Visualization, C.V., L.M., S.C.; Writing, L.M., C.V., S.C. All authors have read and agreed to the published version of the manuscript.  
This research is co-financed by Greece and the European Union (European Social Fund- ESF) through the Operational Programme <<Human Resources Development, Education and Lifelong Learning>> in the context of the project \"Reinforcement of Postdoctoral Researchers--2 nd Cycle\" (MIS-5033021), implemented by the State Scholarships Foundation (IKY). This research is funded by PHENIKAA University under grant number 03.2019.02. The research leading to these results has received funding from the Fond for Scientific Research Vlaanderen (FWO) projects G028015N and G090117N and the FNRS-FWO under Excellence of Science (EOS) Project no 30468160 \"Structured low-rank matrix / tensor approximation: numerical optimization-based algorithms and applications\".  
The authors would like to thank the anonymous reviewers for their valuable comments that have improved the quality of this work.  
The authors declare no conflict of interest.