# Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation

^{1}

^{2}

^{*}

## Abstract

**:**

_{n}

_{+1}= ∓Af

_{NL}(Bx

_{n}) ± Cx

_{n}± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a “unified sigmoidal chaotic map” generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., x

_{n}

_{+1}= ∓f

_{NL}(Bx

_{n}) ± Cx

_{n}, through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01.

## 1. Introduction

## 2. Generic One-Dimensional Sigmoidal Chaotic Maps

#### 2.1. Unification of Generic Sigmoidal Chaotic Map

_{n}is a real variable, f

_{NL}(x

_{n}) is a sigmoidal function, and the parameters A, B, C, and D are real constants. With reference to (1), this paper initially considers a typical sigmoid function, which exhibits S-shaped transfer function characteristics within the range (0, 1) throughout an entire domain (−∞, +∞). In other words, a mathematical model is f(x) = 1/(1 + exp(−x)). Nonetheless, the substitution of the sigmoid function as f

_{NL}(x

_{n}) in (1) could not induce chaos. Therefore, this paper realizes a modified sigmoid function f

_{mod}(x) as follows:

_{ms}(x) is a typical sigmoid function where the function is doubled and shifted down to be (−1, 1). Notice that the nonlinearity in (2) apparently associates to a hyperbolic tangent (tanh) function, i.e.,

#### 2.2. Simplification of Generic Sigmoidal Chaotic Map

_{NL}(x) with S-shaped transfer function characteristics. With respect to the mathematical aspects, the cases NM

_{1}, NM

_{2}, and NM

_{3}are based on inverse trigonometric properties. Meanwhile, the case NM

_{4}is a special function in the form of an integral, which is originally derived from a Gaussian function, while NM

_{5}and NM

_{6}are special differentiable algebraic functions.

_{4}has a range in the y-axis in the region (−1, 1), which closely resembles nonlinearity in a unified sigmoidal chaotic map, whereas the range of NM

_{2}appears to be (−∞, +∞). The ranges of the four remaining cases are limited at certain specific levels. This phenomenon implies that the S-shaped nonlinearity that plays an important role in inducing chaos occurs in a short domain of approximately (−2, 2), and, therefore, the parameter B, which was introduced in the generic sigmoidal chaotic map, consequently becomes a significant factor in determining the chaos dynamics.

## 3. Linearization of Simplified Sigmoidal Chaotic Map for Robust Chaos

_{n})| = f’(x

_{n}), is considered. Typically, the discrete time system becomes unstable in the condition of |J(x

_{n})| > 1, while the chaotic map needs to operate under an unstable condition in order to induce the chaos. With reference to (9), the unstable region of the linearized sigmoidal chaotic maps based on the hardtanh function, which is the parameter region where the chaos can occur, is calculated and provides the following result,

_{n}

_{+1}= f(x

_{n}), typically has a point where x* = f(x*) and is considered a fixed point (equilibrium). Table 2 summarizes the fixed points of the linearized sigmoidal chaotic maps based on the hardtanh function in (13) and (14) and the signum function in (15) and (16), all of which appear to have three fixed points. Figure 7 and Figure 8 show the characteristics of the chaotic waveforms in the time domain as well as the histogram and cobweb plots at specific parameters, which were arbitrarily selected with regard to the chaotic regime, as seen in the bifurcation structure in Figure 5 and Figure 6. The characteristics of the cobweb plots are associated with the fixed points of the chaotic maps, as shown in Table 2. In the case where the fixed point is 0, it is a globally asymptotically stable point, as in |J(0)| = 0. The stability of the fixed point appears in the cobweb plot, where the inward spiral corresponds to the attraction of the stable fixed point, while the outward spiral corresponds to the repelling of the unstable fixed point. The complex closed loops in the cobweb represent a high period of orbit, which indicates an infinite number of non-repeating values. The cobweb plots also relate to the boundary values of x

_{n}

_{+l}, which depend upon the nonlinear term of the chaotic map, and for both cases of the linearized sigmoidal chaotic map in (9) and (10), the values of x

_{n}

_{+l}fall into the region (−1, 1).

## 4. True Random Bit Generation Based on the Proposed Linearized Sigmoidal Chaotic Map

#### 4.1. Random Bit Generator

#### 4.1.1. Entropy Source

_{n}of the signum function, as a result of specifying the V−, which is a reference voltage of the comparator, as 0. In order for the comparator to perform as the signum function, the circuit is supplied with +1V and −1V as +Vcc and –Vcc, respectively.

_{n}is amplified by the non-inverting operational amplifier gain, as V

_{out}= V

_{in}(1 + R

_{2}/R

_{1}), where R

_{3}, R

_{4}, R

_{5}, and R

_{6}are set to be equal. The subtraction of the output of the comparator from the amplified input ax

_{n}results in the output of the chaotic map, x

_{n}

_{+1}.

#### 4.1.2. Entropy Harvester

#### 4.1.3. Post-Processor

#### 4.2. Randomness Performance Evaluation

#### 4.2.1. NIST SP800-22 Test Suite

#### 4.2.2. TestU01

^{20}, 2

^{25}, and 2

^{30}bits were generated from the proposed TRBG. The bit sequences were applied to the batteries Rabbit, Alphabit, and BlockAlphabit to evaluate the randomness. Each battery contains a different number of test. The Alphabit contains 17 statistical tests while the BlockAlphabit applies the Alphabit repeatedly to the reordered bits with the 6 different blocks sizes which are 1, 2, 4, 8, 16, and 32. In other words, the BlockAlphabit contains a total number of 17 × 6 = 102 statistical tests. The Rabbit applies 38, 39, and 40 test to the bit sequence with lengths 2

^{20}, 2

^{25}, and 2

^{30}bits, respectively. The results of the TestU01 are presented in Table 4. The proposed TRBG can pass all the tests.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Majumdar, M.; Mitra, T. Robust ergodic chaos in discounted dynamic optimization models. Econ. Theory
**1994**, 4, 677–688. [Google Scholar] [CrossRef] - Dogaru, R.; Murgan, A.T.; Ortmann, S.; Glesner, M. Searching for robust chaos in discrete time neural networks using weight space exploration. Int. Conf. Neural Netw.
**1996**, 2, 688–693. [Google Scholar] - Banerjee, S.; Yorke, J.A.; Grebogi, C. Robust chaos. Phys. Rev. Lett.
**1998**, 80, 3049–3052. [Google Scholar] [CrossRef] - Shukla, P.K.; Khare, A.; Rizvi, M.A.; Stalin, S.; Kumar, S. Applied cryptography using chaos function for fast digital logic-based systems in ubiquitous computing. Entropy
**2015**, 17, 1387–1410. [Google Scholar] [CrossRef] - Wang, X.Y.; Zhang, Y.Q.; Bao, X.M. A colour image encryption scheme using permutation-substitution based on chaos. Entropy
**2015**, 17, 3877–3897. [Google Scholar] [CrossRef] - Fallahi, K.; Leung, H. A chaos secure communication scheme based on multiplication modulation. Commun. Nonlinear Sci. Numer. Simul.
**2010**, 15, 368–383. [Google Scholar] [CrossRef] - Miliou, A.N.; Antoniades, I.P.; Stavrinides, S.G.; Anagnostopoulos, A.N. Secure communication by chaotic synchronization: Robustness under noisy conditions. Nonlinear Anal. Real World Appl.
**2007**, 8, 1003–1012. [Google Scholar] [CrossRef] - Xu, G.; Shekofteh, Y.; Akgül, A.; Li, C.; Panahi, S. A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation. Entropy
**2018**, 20, 86. [Google Scholar] [CrossRef] - San-Um, W.; Srichavengsup, W. A robust hash function using cross-coupled chaotic maps with absolute-valued sinusoidal nonlinearity. Int. J. Adv. Comput. Sci. Appl.
**2016**. [Google Scholar] [CrossRef] - Andrecut, M.; Ali, M. On the occurrence of robust chaos in a smooth system. Mod. Phys. Lett.
**2001**, 15, 391–395. [Google Scholar] [CrossRef] - Andrecut, M.; Ali, M. Robust chaos in a smooth system. Int. J. Mod. Phys.
**2001**, 15, 177–189. [Google Scholar] [CrossRef] - Pérez, G. Robust chaos in polynomial unimodal maps. Int. J. Bifurc. Chaos
**2004**, 14, 2431–2437. [Google Scholar] [CrossRef] - Alvarez-Llamoza, O.; Cosenza, M.G.; Ponce, G.A. Critical behavior of the Lyapunov exponent in type-III intermittency. Chaos Solitons Fractals
**2008**, 36, 150–156. [Google Scholar] [CrossRef] - Aguirregabiria, J.M. Robust chaos with prescribed natural invariant measure and Lyapunov exponent. arXiv, 2009; arXiv:0907.3790. [Google Scholar]
- Hrusak, J.; Mayer, D.; Stork, M. Structural synthesis of state space energy based adaptive controller for robust chaos-generating systems of arbitrary finite order. In Proceedings of the 2012 International Conference on Applied Electronics (AE), Pilsen, Czech Republic, 5–7 September 2012; pp. 107–110. [Google Scholar]
- Nikulchev, E. Robust chaos generation on the basis of symmetry violations in attractors. In Proceedings of the 2nd International Conference on Emission Electronics (ICEE), St. Petersburg, Russia, 30 June–4 July 2014; pp. 1–3. [Google Scholar]
- Nikulchev, E.V. Generation of robust chaos in the invariant centre manifold. In Proceedings of the 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP), St. Petersburg, Russia, 5–9 October 2015; 2015; pp. 290–291. [Google Scholar]
- Elhadj, Z.; Sprott, J.C. Is A Unifying Chaotic Dynamical System Possible? Int. J. Open Probl. Comput. Sci. Math.
**2012**, 5, 75–78. [Google Scholar] [CrossRef] - Elhadj, Z.; Sprott, J.C. The unified chaotic system describing the Lorenz and Chua systems. Facta Univ. Series Electron. Energ.
**2010**, 23, 345–355. [Google Scholar] [CrossRef] - Elhadj, Z.; Sprott, J.C. A unified piecewise smooth chaotic mapping that contains the Hénon and the Lozi systems. Annu. Rev. Chaos Theory Bifurc. Dyn. Syst.
**2011**, 1, 50–60. [Google Scholar] - Marwan, N.; Romano, M.C.; Thiel, M.; Kurths, J. Recurrence plots for the analysis of complex systems. Phys. Rep.
**2007**, 438, 237–329. [Google Scholar] [CrossRef] - Nejati, H.; Beirami, A.; Ali, W.H. Discrete-time chaotic-map truly random number generators: Design, implementation, and variability analysis of the zigzag map. Analog. Integr. Circuits Signal Process.
**2012**, 73, 363–374. [Google Scholar] [CrossRef] - Callegari, S.; Rovatti, R.; Setti, G. Embeddable ADC-based true random number generator for cryptographic applications exploiting nonlinear signal processing and chaos. IEEE Trans. Signal Process.
**2005**, 53, 793–805. [Google Scholar] [CrossRef] - Liu, L.; Miao, S.; Hu, H.; Deng, Y. Pseudorandom bit generator based on non-stationary logistic maps. IET Inf. Secur.
**2016**, 10, 87–94. [Google Scholar] [CrossRef] - Elhadj, Z.; Sprott, J.C. On the robustness of chaos in dynamical systems: Theories and applications. Front. Phys. China
**2008**, 3, 195–204. [Google Scholar] [CrossRef] - Pareschi, F.; Rovatti, R.; Setti, G. Simple and effective post-processing stage for random stream generated by a chaos-based RNG. In Proceedings of the 2006 International Symposium on Nonlinear Theory and its Applications (NOLTA2006), Bologna, Italy, 11–14 September 2006. [Google Scholar]
- Rukin, A.; Soto, J.; Nechvatal, J.; Smid, M.; Barker, E. A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications; National Institute of Standards and Technology: Gaithersburg, MD, USA, 2001.
- L'Ecuyer, P.; Simard, R. TestU01: AC library for empirical testing of random number generators. ACM Trans. Math. Softw
**2007**, 33, 22. [Google Scholar] [CrossRef]

**Figure 1.**Plots of a bifurcation structure of parameters C versus B of the unified sigmoidal chaotic map in (5), where the heat diagram indicates a positive Lyapunov exponent. LE = Lyapunov exponent.

**Figure 2.**Characteristics of chaotic waveforms in time domain and plots of histogram, cobweb, and frequency spectrum using periodogram at specific parameters B = 75 and C = 1.9; (

**a**–

**d**) characteristics of Equation (5), (

**e**–

**h**) characteristics of Equation (6).

**Figure 3.**Plots of transfer function characteristics of the nonlinear functions of the cases NM

_{1}to NM

_{6}.

**Figure 4.**The plots of unstable and chaos regions with reference to (17), where the regions in grey and blue represent the unstable region and the chaos region, respectively.

**Figure 5.**Plots of a bifurcation structure of parameter C versus B of the hardtanh-based linearized sigmoidal chaotic map in (13), where the heat diagram indicates a positive Lyapunov exponent.

**Figure 6.**Plots of a bifurcation structure of parameters C versus B of the signum-based linearized sigmoidal chaotic map in (15), where the heat diagram indicates a positive Lyapunov exponent.

**Figure 7.**Characteristics of chaotic waveforms in time domain and plots of histogram, cobweb, and frequency spectrum using periodogram at specific parameters B = 15 and C = 1.9; (

**a**–

**d**) characteristics of Equation (13), (

**e**–

**h**) characteristics of Equation (14).

**Figure 8.**Characteristics of chaotic waveforms in time domain and plots of histogram, cobweb, and frequency spectrum using periodogram at specific parameter B = 1 and C = 1.9; (

**a**–

**d**) characteristics of Equation (15), (

**e**–

**h**) characteristics of Equation (16).

**Figure 9.**Plots of Bifurcation diagram and Lyapunov exponents (LEs) of chaotic maps at specific parameter B = 75; (

**a**,

**d**) the unified sigmoidal chaotic map in (5), (

**b**,

**e**) the hardtanh-based linearized sigmoidal chaotic map in (13), (

**c**,

**f**) signum-based linearized sigmoidal chaotic map in (15).

**Figure 10.**Recurrence plots of the signum-based linearized sigmoidal chaotic map in (15) for two different dynamic regimes, at specific parameter B = 1; (

**a**) periodic regime: parameter C = 0.5, (

**b**) chaotic regime: parameter C = 1.9.

**Figure 11.**Proposed true random bit generator based on the signum-based linearized sigmoidal chaotic map.

**Figure 12.**Circuit realizing the chaotic map with reference to the signum-based linearized sigmoidal chaotic map in (15).

**Figure 13.**Plots of entropy versus threshold T and parameter C of the signum-based linearized sigmoidal chaotic map in (15).

**Table 1.**Summary of six simplified sigmoidal chaotic maps involving nonlinear functions f

_{NL}(x) with S-shaped transfer function characteristics.

Cases | Descriptions | f_{NL}(x) with No Parameters | Chaotic Maps |
---|---|---|---|

NM_{1} | Inverse Tangent Function | ${f}_{\mathrm{NL}1}(x)=\mathrm{tan}{}^{-1}\left(x\right)$ | ${x}_{n+1}=\mp \mathrm{tan}{}^{-1}(B{x}_{n})\pm C{x}_{n}$ |

NM_{2} | Inverse Hyperbolic Sine Function | ${f}_{\mathrm{NL}2}(x)=\mathrm{sin}\mathrm{h}{}^{-1}\left(x\right)$ | ${x}_{n+1}=\mp \mathrm{sin}\mathrm{h}{}^{-1}(B{x}_{n})\pm C{x}_{n}$ |

NM_{3} | Gudermannian Function | ${f}_{\mathrm{NL}3}(x)=\mathrm{tan}{}^{-1}\left(\mathrm{sinh}\text{\hspace{0.17em}}\left(x\right)\right)$ | ${x}_{n+1}=\mp \mathrm{tan}{}^{-1}(\mathrm{sin}\mathrm{h}\text{\hspace{0.17em}}(B{x}_{n}))\pm C{x}_{n}$ |

NM_{4} | Error Function | ${f}_{\mathrm{NL}4}(x)=\frac{2}{\sqrt{\pi}}{\displaystyle {\int}_{0}^{x}{e}^{-{t}^{2}}dt}$ | ${x}_{n+1}=\mp \frac{2}{\sqrt{\pi}}{\displaystyle {\int}_{0}^{Bx}{e}^{-{t}^{2}}dt}\pm C{x}_{n}$ |

NM_{5} | Soft Signum Function | ${f}_{\mathrm{NL}5}(x)=\frac{x}{1+\left|x\right|}$ | ${x}_{n+1}=\mp \frac{B{x}_{n}}{1+\left|B{x}_{n}\right|}\pm C{x}_{n}$ |

NM_{6} | Specific Algebraic Function | ${f}_{\mathrm{NL}6}(x)=\frac{x}{\sqrt{1+{x}^{2}}}$ | ${x}_{n+1}=\mp \frac{B{x}_{n}}{\sqrt{1+{(B{x}_{n})}^{2}}}\pm C{x}_{n}$ |

Chaotic Map Equations | x* = f(x*) | Fixed Points x* |
---|---|---|

(10) | $x*=\mathrm{hardtanh}\text{}(Bx*)-Cx*$ | $0,\text{}\frac{1}{C-1}$ and $-\frac{1}{C-1}$ |

(11) | $x*=-\mathrm{hardtanh}\text{}(Bx*)+Cx*$ | $0,\text{}\frac{1}{C+1}$ and $-\frac{1}{C+1}$ |

(12) | $x*=\mathrm{sgn}\text{}(Bx*)-Cx*$ | $0,\text{}\frac{1}{C-1}$ and $-\frac{1}{C-1}$ |

(13) | $x*=-\mathrm{sgn}\text{}(Bx*)+Cx*$ | $0,\text{}\frac{1}{C+1}$ and $-\frac{1}{C+1}$ |

Test Methods | P-Value | Proportion | Result |
---|---|---|---|

Frequency (monobit) | 0.7981 | 0.99 | Pass |

Block Frequency | 0.5544 | 0.99 | Pass |

Runs | 0.6163 | 1.00 | Pass |

Longest Run | 0.7399 | 1.00 | Pass |

Binary Matrix Rank | 0.2133 | 1.00 | Pass |

Discrete Fourier Transform | 0.7791 | 1.00 | Pass |

Non-overlapping Template Matching | 0.4980 | 0.99 | Pass |

Overlapping Template Matching | 0.9114 | 0.98 | Pass |

Universal Statistical | 0.7597 | 0.99 | Pass |

Linear Complexity | 0.6579 | 0.99 | Pass |

Serial | 0.4983 | 0.98 | Pass |

Approximate Entropy | 0.3669 | 1.00 | Pass |

Cumulative Sums | 0.5139 | 0.99 | Pass |

Random Excursions | 0.3322 | 0.98 | Pass |

Random Excursions Variant | 0.3384 | 0.99 | Pass |

Random Bit Generator | Test Batteries | ||
---|---|---|---|

Rabbit | Alphabit | BlockAlphabit | |

2^{20} bits | |||

Proposed TRBG | 38/38 | 17/17 | 102/102 |

2^{25} bits | |||

Proposed TRBG | 39/39 | 17/17 | 102/102 |

2^{30} bits | |||

Proposed TRBG | 40/40 | 17/17 | 102/102 |

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**MDPI and ACS Style**

Jiteurtragool, N.; Masayoshi, T.; San-Um, W. Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation. *Entropy* **2018**, *20*, 136.
https://doi.org/10.3390/e20020136

**AMA Style**

Jiteurtragool N, Masayoshi T, San-Um W. Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation. *Entropy*. 2018; 20(2):136.
https://doi.org/10.3390/e20020136

**Chicago/Turabian Style**

Jiteurtragool, Nattagit, Tachibana Masayoshi, and Wimol San-Um. 2018. "Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation" *Entropy* 20, no. 2: 136.
https://doi.org/10.3390/e20020136