**5. Conclusions**

In this article, the **IEGBNDS** algorithm via Newton-Raphson's method and general Bischi-Naimzadah duopoly system (**GBNDS**) has been suggested. Newton-Raphson's method has been used for shuffling the rows/columns of the plain image. **GBNDS** has been used to producing chaotic sequences to diffusion phase of image encryption algorithm. The extracted chaotic sequences from the **GBNDS** is extremely random based on the NIST statistical tests. Many security experiments are applied to evaluate the efficiency of our algorithm. The **IEGBNDS** algorithm has a large key space (10<sup>14</sup>(*k*+<sup>1</sup>) + <sup>10</sup>168(>>2<sup>100</sup>), the histograms of the generated cipher images are close to the uniform distributions, all entropy values for the cipher images based on **IEGBNDS** algorithm are close to 8, all correlation coefficient values for the cipher images are close to zero. The **IEGBNDS** algorithm outperforms some recent algorithms at least in one of the two measures, highly sensitive to small changes of the secret key, can be robust against the noise and data loss attacks, and can hold out against the plaintext attacks. In comparison to several recent algorithms, the **IEGBNDS** algorithm has a small running time. NIST statistical tests for 100 cipher images by the **IEGBNDS** algorithm are performed and all tests are passed. Finally, quantum image encryption algorithm based on **GBNDS** will be designed in the future to increase the security of the current algorithm.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing not applicable.

**Acknowledgments:** I deeply thank Shehzad Ahmed for his proof reading the paper.

**Conflicts of Interest:** The author declares no conflict of interest.
