# Self-Parameterized Chaotic Map for Low-Cost Robust Chaos

^{*}

## Abstract

**:**

## 1. Introduction

- We present the idea of self-parameterization in more detail.
- At first, similar to the conference paper, the scheme is demonstrated in the case of three ideal mathematical maps: Logistic, Sine, and Tent maps.
- A general design methodology, in the light of stability analysis, was added to this paper.
- Similar to the conference paper, we present a design for a digitized SPM that can be implemented in FPGA.
- Then, we show the hardware-efficient CMOS-based designs for the analog implementation of SPM. In this paper, we present three different topologies of maps, and introduce the corresponding low transistor-count transformation circuits.
- The chaotic performance of the proposed designs was analyzed with different entropy metrics, along with one additional entropy metric, in this paper.
- We added an application of the proposed scheme in this paper. The application was demonstrated in a random number generator design. The cryptographic applicability of the random number generator was verified with an established statistical tool.

## 2. Proposed Scheme

## 3. Analytical Function-Based Maps

#### 3.1. Design Methodology

#### 3.2. Chaotic Performance Evaluation

#### 3.2.1. The Lyapunov Exponent

#### 3.2.2. Correlation Coefficient

#### 3.2.3. Correlation Dimension

#### 3.3. Approximate Entropy

## 4. Design for FPGA Implementation

## 5. CMOS Implementation

## 6. Hardware Efficiency

## 7. Application

#### 7.1. Design of RNG

#### 7.2. NIST

#### 7.3. Hardware Considerations

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**(

**a**–

**c**) Bifurcation plots of the seed maps; (

**d**–

**f**) slope of the transfer curve at the intersection between the transfer curves and ${x}_{n+1}={x}_{n}$-line.

**Figure 5.**(

**a**–

**c**) Bifurcation plots of the cascaded pairs of the seed maps; (

**d**–

**f**) slope of the cascaded transfer curve at the intersection between the transfer curves and ${x}_{n+1}={x}_{n}$-line.

**Figure 7.**Transfer curves for self-parameterized maps (SPMs). The transformation parameters for the SPMs are: Logistic: $B=0.975,M=0.01$; Tent: $B=0.74,M=0.125$; Sine: $B=0.96,M=0.01$.

**Figure 8.**Slopes of the (self-parameterized map) SPMs’ transfer curves at the intersections between the transfer curves and ${x}_{n+1}={x}_{n}$-line: (

**a**–

**c**) single map; (

**d**–

**f**) cascaded pairs.

**Figure 9.**Bifurcation plots of the self-parameterized maps, generated from ideal seed maps—Logistic, Tent, and Sine.

**Figure 10.**Lyapunov Exponent (LE) comparison of seed maps (SM) to corresponding self-parameterized maps (SPM).

**Figure 11.**Comparison of seed maps (SM) to corresponding self-parameterized maps (SPM), in terms of the Correlation Coefficient measurement, by varying the initial state ($C{C}_{{x}_{0}}$), and by varying the control parameter ($C{C}_{C}$).

**Figure 12.**Comparison of seed maps (SM) to corresponding self-parameterized maps (SPM), in terms of the Correlation Dimension (CD) measurement.

**Figure 13.**Comparison of seed maps (SM) to corresponding self-parameterized maps (SPMs), in terms of the Approximate Entropy (AE) measurement.

**Figure 15.**Comparison between the simulated trajectory from the digital implementation (FPGA) and the analytical model (MATLAB).

**Figure 16.**Comparison between the results from the digital implementation (FPGA) and the analytical model (MATLAB): (

**a**) Lyapunov Exponent (LE) (

**b**); Correlation Dimension (CD).

**Figure 22.**Slopes of the self-parameterized map (SPM) transfer curves at the intersections between the transfer curves and ${x}_{n+1}={x}_{n}$-line: (

**a**–

**c**) single SPM; (

**d**–

**f**) cascade of two SPMs.

**Figure 26.**Lyapunov Exponent (LE) comparison between CMOS-based implementations of seed maps (SMs) and corresponding self-parameterized maps (SPMs).

**Figure 27.**Comparison between the CMOS-based seed maps (SM) and the corresponding self-parameterized maps (SPM), in terms of the correlation coefficient measurement by varying the initial state ($C{C}_{{x}_{0}}$) and by varying the control parameter ($C{C}_{C}$).

**Figure 28.**Comparison between the CMOS-based implementations of seed maps (SM) and the corresponding self-parameterized maps (SPM), in terms of the correlation dimension (CD) measurement.

**Figure 29.**Comparison between the CMOS-based implementations of seed maps (SM) and the corresponding self-parameterized maps (SPM), in terms of the approximate entropy (AE) measurement.

**Figure 31.**Results of the NIST test. Averages of multiple tests were used for the cases of ’*’-marked test names.

Map | Analytical Expression | Output Range | Control Parameter Range | Chaotic Range |
---|---|---|---|---|

Name | $\left({\mathit{f}}_{\mathit{SM}}\right)$ | $[{\mathit{x}}_{\mathit{nl}}$, ${\mathit{x}}_{\mathit{nh}}]$ | $[{\mathit{C}}_{\mathit{l}}$, ${\mathit{C}}_{\mathit{h}}]$ | $[{\mathit{C}}_{\mathit{l}}^{\prime}$, ${\mathit{C}}_{\mathit{h}}^{\prime}]$ |

Logistic | ${x}_{n+1}=4C{x}_{n}(1-{x}_{n})$ | [0, 1] | [0, 1] | [0.9, 1] |

Sine | ${x}_{n+1}=C\mathrm{sin}\left(\pi {x}_{n}\right)$ | [0, 1] | [0, 1] | [0.87, 1] |

Tent | ${x}_{n+1}=\left\{\begin{array}{c}\hfill 2C{x}_{n}\phantom{\rule{30.0pt}{0ex}};{x}_{n}<0.5\hfill \\ \hfill 2C(1-{x}_{n})\phantom{\rule{3.00003pt}{0ex}};{x}_{n}\ge 0.5\hfill \end{array}\right.$ | [0, 1] | [0, 1] | [0.52, 0.99] |

Map | Selected Parameter Values | Analytical Expression |
---|---|---|

Name | M, B | ${\mathit{f}}_{\mathit{SPM}}(\mathit{C},{\mathit{x}}_{\mathit{n}})$ |

SPM-Logistic | M = 0.01, B = 0.975 | ${x}_{n+1}=[0.04({x}_{n}+C)+3.9]{x}_{n}(1-{x}_{n})$ |

SPM-Sine | M = 0.01, B = 0.96 | ${x}_{n+1}=[0.01({x}_{n}+C)+0.96]\mathrm{sin}\left(\pi {x}_{n}\right)$ |

SPM-Tent | M = 0.125, B = 0.74 | ${x}_{n+1}=\left\{\begin{array}{c}\hfill [0.25({x}_{n}+C)+1.48]{x}_{n}\phantom{\rule{30.0pt}{0ex}};{x}_{n}<0.5\hfill \\ \hfill [0.25({x}_{n}+C)+1.48](1-{x}_{n})\phantom{\rule{3.00003pt}{0ex}};{x}_{n}\ge 0.5\hfill \end{array}\right.$ |

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## Share and Cite

**MDPI and ACS Style**

Paul, P.S.; Dhungel, A.; Sadia, M.; Hossain, M.R.; Hasan, M.S.
Self-Parameterized Chaotic Map for Low-Cost Robust Chaos. *J. Low Power Electron. Appl.* **2023**, *13*, 18.
https://doi.org/10.3390/jlpea13010018

**AMA Style**

Paul PS, Dhungel A, Sadia M, Hossain MR, Hasan MS.
Self-Parameterized Chaotic Map for Low-Cost Robust Chaos. *Journal of Low Power Electronics and Applications*. 2023; 13(1):18.
https://doi.org/10.3390/jlpea13010018

**Chicago/Turabian Style**

Paul, Partha Sarathi, Anurag Dhungel, Maisha Sadia, Md Razuan Hossain, and Md Sakib Hasan.
2023. "Self-Parameterized Chaotic Map for Low-Cost Robust Chaos" *Journal of Low Power Electronics and Applications* 13, no. 1: 18.
https://doi.org/10.3390/jlpea13010018