# An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Fractional-Order Memristive Chaotic System

## 3. A PRNG and the Encryption Scheme

#### 3.1. Encryption Scheme

#### 3.2. Performance Evaluation

#### 3.2.1. Statistical Tests

#### 3.2.2. Key Space and Key Sensitivity

#### 3.2.3. Resistance to Differential Attacks

## 4. FPGA Implementation and Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Performance Measure | Equation |
---|---|

Correlation coefficient | $\rho =\frac{cov(x,y)}{\sqrt{\left(D\right(x\left)\right)}\sqrt{\left(D\right(y\left)\right)}}$, where $cov(x,y)=\frac{1}{S}{\sum}_{i=1}^{S}\left({x}_{i}-\frac{1}{s}{\sum}_{j=1}^{S}{x}_{j}\right)\left({y}_{i}-\frac{1}{S}{\sum}_{j=1}^{S}{y}_{j}\right)$, |

$D\left(x\right)=\frac{1}{s}{\sum}_{i=1}^{S}{\left({x}_{i}-\frac{1}{S}{\sum}_{j=1}^{S}{x}_{j}\right)}^{2}$, $S=M$ (height) $\times N($ width). | |

Mean squared error | MSE $=\frac{1}{M\times N}{\sum}_{i=1}^{N}{\sum}_{j=1}^{M}{(P(i,j)-D(i,j))}^{2}$, where $P(i,j),D(i,j)$ are the original $\mathrm{and}$ |

incorrectly decrypted image pixels. | |

Entropy | $E=-{\sum}_{i=1}^{{2}^{8}}p\left({s}_{i}\right){log}_{2}p\left({s}_{i}\right)$, where $p\left({s}_{i}\right)$ is the probability of symbol ${s}_{i}$. |

Number of pixels | |

change rate | $\mathrm{NPCR}=\frac{1}{N\times M}{\sum}_{i=1}^{N}{\sum}_{j=1}^{M}D(i,j)\times 100$, |

Unified average | |

changing intensity | $\mathrm{UACI}=\frac{1}{M\times N}{\sum}_{i=1}^{N}{\sum}_{j=1}^{M}\left|\frac{{C}_{1}(i,j)-{C}_{2}(i,j)}{255}\right|\times 100$, |

where $D(i,j)=\left\{\begin{array}{cc}1,\hfill & {C}_{1}(i,j)\ne {C}_{2}(i,j)\hfill \\ 0,\hfill & {C}_{1}(i,j)={C}_{2}(i,j)\hfill \end{array},{C}_{1}\right.$ is the ciphered pixel and | |

${C}_{2}$ is the ciphered pixel corresponding to a slightly modified original image. |

Image | Horizontal | Vertical | Diagonal | ||||||
---|---|---|---|---|---|---|---|---|---|

R | G | B | R | G | B | R | G | B | |

Baboon | $-0.0005$ | $0.0021$ | $0.0024$ | $-0.0023$ | $-0.0005$ | $-0.0019$ | $0.0001$ | $0.0033$ | $0.0016$ |

Peppers | $-0.0015$ | $0.0049$ | $0.0035$ | $0.0008$ | $-0.0016$ | $-0.0013$ | $-0.0013$ | $0.0016$ | $-0.0005$ |

Barb | $0.0021$ | $0.0042$ | $-0.0027$ | $0.0016$ | $-0.0028$ | $0.00058$ | $-0.0008$ | $-0.0015$ | $0.0001$ |

Lake | $-0.0004$ | $-0.0002$ | $0.0006$ | $-0.0005$ | $0.0003$ | $-0.0008$ | $0.0007$ | $0.0011$ | $-0.0011$ |

Baboon | Peppers | Barb | Lake | Baboon [34] | Peppers [34] | ||
---|---|---|---|---|---|---|---|

Entropy | $7.9998$ | $7.9997$ | $7.9998$ | $7.9998$ | $7.9998$ | $7.9998$ | |

MSE | 8609 | 10151 | 8540 | 10115 | 8710 | 8798 | |

Key sens. | Entropy | $7.9998$ | $7.9997$ | $7.9997$ | $7.9997$ | $7.9992$ | $7.9991$ |

MSE | 8502 | 9935 | 8408 | 9950 | 8701 | 8034 | |

DA | NPCR (%) | $99.5641$ | $99.6005$ | $99.6017$ | $99.5873$ | $99.6101$ | $99.6096$ |

UACI (%) | $33.4394$ | $33.4793$ | $33.5018$ | $33.4418$ | $33.4582$ | $33.4593$ |

Test | PRNG | Baboon | Peppers | Barb | Lake | |||||
---|---|---|---|---|---|---|---|---|---|---|

PV | PP | PV | PP | PV | PP | PV | PP | PV | PP | |

Frequency | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | $0.958$ | ✔ | 1 |

Block frequency | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | $0.958$ | ✔ | $0.917$ |

Cumulative sums | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 |

Runs | ✔ | 1 | ✔ | $0.958$ | ✔ | 1 | ✔ | 1 | ✔ | 1 |

Longest run | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | $0.917$ | ✔ | 1 |

Rank | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 |

FFT | ✔ | 1 | ✔ | $0.958$ | ✔ | 1 | ✔ | 1 | ✔ | $0.917$ |

Non-overlapping template | ✔ | $0.989$ | ✔ | $0.990$ | ✔ | $0.989$ | ✔ | $0.988$ | ✔ | $0.988$ |

Overlapping template | ✔ | 1 | ✔ | $0.917$ | ✔ | $0.958$ | ✔ | 1 | ✔ | 1 |

Universal | ✔ | $0.958$ | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | $0.958$ |

Approximate entropy | ✔ | $0.958$ | ✔ | $0.958$ | ✔ | $0.958$ | ✔ | 1 | ✔ | $0.958$ |

Random excursions | ✔ | $0.991$ | ✔ | 1 | ✔ | 1 | ✔ | $0.981$ | ✔ | 1 |

Random excursions variant | ✔ | $0.972$ | ✔ | 1 | ✔ | 1 | ✔ | $0.970$ | ✔ | $0.984$ |

Serial | ✔ | $0.979$ | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 |

Linear complexity | ✔ | $0.917$ | ✔ | 1 | ✔ | 1 | ✔ | 1 | ✔ | 1 |

Final result | Passed | Passed | Passed | Passed | Passed |

Total Slices | Total Slice LUTs | Total Slice Regs | DSPs | Max Freq (MHz) | Throughput (Gbit/s) | Order | Params. | |
---|---|---|---|---|---|---|---|---|

Proposed | 6269 | 23,929 | 4599 | 144 | 12.368 | 0.396 | Fractional | 8 |

[30] | 951 | 3440 | 374 | 84 | 14.009 | 0.4483 | Integer | 4 |

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**MDPI and ACS Style**

Mohamed, S.M.; Sayed, W.S.; Madian, A.H.; Radwan, A.G.; Said, L.A. An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System. *Electronics* **2023**, *12*, 1219.
https://doi.org/10.3390/electronics12051219

**AMA Style**

Mohamed SM, Sayed WS, Madian AH, Radwan AG, Said LA. An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System. *Electronics*. 2023; 12(5):1219.
https://doi.org/10.3390/electronics12051219

**Chicago/Turabian Style**

Mohamed, Sara M., Wafaa S. Sayed, Ahmed H. Madian, Ahmed G. Radwan, and Lobna A. Said. 2023. "An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System" *Electronics* 12, no. 5: 1219.
https://doi.org/10.3390/electronics12051219