# Memristive Structure-Based Chaotic System for PRNG

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of Mathematical Model and Solutions Simulation

#### 2.1. Memristive Structure-Based Chaotic System

#### 2.2. Solutions Simulation

## 3. Generating of Pseudorandom Sequences

#### 3.1. Time Series Balance Property

#### 3.2. Design of CPRNG

#### 3.3. FPGA-Implementation

#### 3.4. Testing of CPRNG Based on One System

- 1 bit defined the sign;
- 5 bits were used for the integer part;
- 26 bits were used for the fractional part.

## 4. Coupled Memristive Structure-Based Chaotic Circuits for PRNG

#### 4.1. Coupled Chaotic Systems without Synchronization

#### 4.2. Testing of CPRNG Based on Coupled System

## 5. Application

#### 5.1. Image Encryption

#### 5.2. Security Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The dependence of the Lyapunov exponents for (

**a**) $C=1F$, $\beta =1.52$, $\alpha =0.6$; (

**b**) $L=3H$, $\beta =1.52$, $\alpha =0.6$; (

**c**) $C=1F$, $L=3H$, $\alpha =0.6$.

**Figure 3.**Chaotic attractor of $(x,y)$-plane for system (2). (

**a**,

**c**) for Euler’s method; (

**b**,

**d**) for Runge–Kutta’s method. (

**a**,

**b**) are obtained for $\Delta t=0.001$; (

**c**,

**d**) are obtained for $\Delta t=0.01$.

**Figure 4.**Chaotic attractor obtained by emulator in [35].

**Figure 5.**Histograms of distribution of $(x,y)$-plane for system (2). (

**a**,

**c**) for Euler’s method; (

**b**,

**d**) for Runge–Kutta’s method. (

**a**,

**b**) are obtained for $\Delta t=0.001$; (

**c**,

**d**) are obtained for $\Delta t=0.01$.

**Figure 6.**The balance of bits generated by system (3) that is implemented by a Simulink-model using fixed-point arithmetic Q6.26.

**Figure 7.**The balance of bits in the binary representation of x, y, and z, which are generated by the system (4) implemented as the Simulink model with floating-point double-precision arithmetic.

**Figure 10.**Captured phase portrait of (2) obtained on FPGA.

**Figure 17.**Image encryption: (

**a**) original image, (

**b**) encrypted image, (

**c**) decrypted image, (

**d**) decrypted image by changing the sub key ${x}_{0}$ on ${2}^{-35}$.

**Figure 20.**Histogram of encrypted image by changing sub key $x\left(0\right)$ on ${2}^{-35}$ of (

**a**) red, (

**b**) green, and (

**c**) blue components.

Euler’s Method, FPGA Implementation, 32-bits Fixed-Point Arithmetic | Runge–Kutta Fourth Order Method, Matlab-Simulink Implementation, Double-Precision Floating-Point Arithmetic | |||
---|---|---|---|---|

Test | p-Value/Proportion | Status | p-Value/Proportion | Status |

FT | 0.504219/0.993 | Pass | 0.733899/0.986 | Pass |

BFT | 0.922855/0.990 | Pass | 0.975644/0.987 | Pass |

CST | 0.514124/0.993 | Pass | 0.305599/0.986 | Pass |

CST | 0.811080/0.992 | Pass | 0.610070/0.987 | Pass |

RuT | 0.757790/0.988 | Pass | 0.723804/0.988 | Pass |

LRuT | 0.128132/0.990 | Pass | 0.969588/0.984 | Pass |

RaT | 0.591409/0.991 | Pass | 0.713641/0.992 | Pass |

FFT | 0.317565/0.990 | Pass | 0.494392/0.988 | Pass |

NOTT | All 148 tests passed | All 148 tests are passed | ||

OTT | 0.589341/0.984 | Pass | 0.142062/0.989 | Pass |

UT | 0.118812/0.986 | Pass | 0.524101/0.991 | Pass |

AET | 0.684890/0.986 | Pass | 0.508172/0.989 | Pass |

RET | All 8 tests passed | All 8 tests are passed | ||

REVT | All 18 tests passed | All 18 tests are passed | ||

ST | 0.192724/0.987 | Pass | 0.686955/0.988 | Pass |

ST | 0.182550/0.984 | Pass | 0.478839/0.993 | Pass |

LCT | 0.522100/0.988 | Pass | 0.729870/0.987 | Pass |

Euler’s Method, FPGA 32-bits Fixed- Point Arithmetic Implementation, Systems with Modified Coupling (14) | Euler’s Method, FPGA 32-bits Fixed- Point Arithmetic Implementation, Systems Coupled through Inductance (15) | |||
---|---|---|---|---|

Test | p-Value/Proportion | Status | p-Value/Proportion | Status |

FT | 0.385543/0.991 | Pass | 0.946308/0.987 | Pass |

BFT | 0.846338/0.990 | Pass | 0.016037/0.995 | Pass |

CST | 0.473064/0.993 | Pass | 0.377007/0.989 | Pass |

CST | 0.136499/0.991 | Pass | 0.904708/0.988 | Pass |

RuT | 0.522100/0.986 | Pass | 0.244236/0.990 | Pass |

LRuT | 0.014961/0.994 | Pass | 0.538182/0.990 | Pass |

RaT | 0.141256/0.989 | Pass | 0.044508/0.989 | Pass |

FFT | 0.486588/0.994 | Pass | 0.603841/0.987 | Pass |

NOTT | All 148 tests are passed | All 148 tests passed | ||

OTT | 0.146982/0.984 | Pass | 0.820143/0.990 | Pass |

UT | 0.266235/0.986 | Pass | 0.257004/0.985 | Pass |

AET | 0.662091/0.991 | Pass | 0.729870/0.985 | Pass |

RET | All 8 tests passed | All 8 tests are passed | ||

REVT | All 18 tests passed | All 18 tests are passed | ||

ST | 0.028244/0.998 | Pass | 0.943242/0.993 | Pass |

ST | 0.358641/0.990 | Pass | 0.607993/0.989 | Pass |

LCT | 0.422638/0.992 | Pass | 0.834308/0.997 | Pass |

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**MDPI and ACS Style**

Haliuk, S.; Krulikovskyi, O.; Vovchuk, D.; Corinto, F.
Memristive Structure-Based Chaotic System for PRNG. *Symmetry* **2022**, *14*, 68.
https://doi.org/10.3390/sym14010068

**AMA Style**

Haliuk S, Krulikovskyi O, Vovchuk D, Corinto F.
Memristive Structure-Based Chaotic System for PRNG. *Symmetry*. 2022; 14(1):68.
https://doi.org/10.3390/sym14010068

**Chicago/Turabian Style**

Haliuk, Serhii, Oleh Krulikovskyi, Dmytro Vovchuk, and Fernando Corinto.
2022. "Memristive Structure-Based Chaotic System for PRNG" *Symmetry* 14, no. 1: 68.
https://doi.org/10.3390/sym14010068