# Designing an Image Encryption Scheme Based on Compressive Sensing and Non-Uniform Quantization for Wireless Visual Sensor Networks

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## Abstract

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## 1. Introduction

## 2. Preliminary Knowledge and Designed OLM

#### 2.1. Compressive Sensing

#### 2.2. Proposed OLM

## 3. The Proposed Cryptosystem

- Measurement:In the proposed scheme, the scene at a sensor node was directly measured by a single pixel camera whose micro-mirror array was controlled by a measurement matrix. Generally speaking, in order to satisfy RIP, most measurement matrices are the Gaussian random matrix and the Bernoulli random matrix. In WVSN with security requirements, the measurement matrix is usually generated by a secret key that should be transmitted; hence, a random matrix is no longer suitable. In 2012, Chen et al. [32] proved that matrices constructed by chaotic sequences perform better. More importantly, this kind of matrix could be generated by few parameters, which helps to shrink the size of the key conspicuously. Furthermore, the Toeplitz-structured chaotic measurement matrix has been verified to be capable of satisfying RIP with high probability [33]. To further shorten the length of the chaotic sequence required, a cyclic-structured chaotic matrix:$$\mathit{C}=\left(\begin{array}{cccc}{t}_{n}& {t}_{n-1}& \cdots & {t}_{1}\\ {t}_{1}& {t}_{n}& \cdots & {t}_{2}\\ \vdots & \vdots & \ddots & \vdots \\ {t}_{m-1}& {t}_{m-2}& \cdots & {t}_{m}\end{array}\right)$$
- Quantization:The quantization process could be divided into two parts: the parameter-variable non-linear transform and the conventional UQ. First, the measurement result $\mathit{y}$ was transformed by a non-linear function:$$\mathit{z}=\frac{1}{1+{e}^{-a(\mathit{y}-oset)}},$$$$a={\displaystyle \frac{ln(1/0.05-1)}{2\delta}}$$
- Confusion and substitution:After that, the process of confusion and substitution [34,35] was appended to increase the security of quantized data. In the confusion process, OLM${}_{2}$ was firstly applied to iterate t times to obtain a chaotic sequence, whose length was the same as $\mathit{Q}$’s. Then, we sorted this sequence by the values to acquire an index sequence $\mathit{I}$ ranging from 1–t. Afterwards, we reordered the elements in $\mathit{Q}$ by sequence $\mathit{I}$ to get confused sequence ${\mathit{Q}}_{\mathit{rc}}$. In the substitution step, the CBC structure was avoided to enhance the anti-interference performance of the proposed system. Chaotic sequence $\mathit{B}$ was utilized to conceal the information by:$$C\left(i\right)={Q}_{rc}\left(i\right)\oplus B\left(i\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i\in \{1,2,...,t\},$$

## 4. Simulation and Analysis

#### 4.1. Security Performance

#### 4.1.1. Ability of Resisting CPA

#### 4.1.2. Key Space and Key Sensitivity

#### 4.1.3. Statistical Histogram

#### 4.1.4. Correlation Coefficients

#### 4.1.5. Differential Analysis

#### 4.1.6. Randomness Analysis

#### 4.1.7. Efficiency Analysis

#### 4.1.8. Information Entropy

#### 4.2. Anti-Jamming Performance

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WVSN | Wireless visual sensor networks |

CS | Compressive sensing |

CPA | Chosen-plaintext attack |

NQ | Non-uniform quantization |

CR | Compression ratio |

LM | Logistic map |

OLM | Optimized logistic map |

CCA | Chosen-ciphertext attack |

CBC | Cipher block chaining |

UQ | Uniform quantization |

RIP | Restricted isometry property |

OMP | Orthogonal matching pursuit |

ROMP | Regularized orthogonal matching pursuit |

CCD | Charge coupled device |

UINT8 | Unsigned integers in eight bits |

DCT | Discrete cosine transformation |

XOR | Exclusive or |

NPCR | The number of pixels change rate |

UACI | The unified average changing intensity |

PSNR | Peak signal-to-noise ratio |

AWGN | Additive white Gaussian noise |

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**Figure 5.**Key sensitivity in the encryption part. (

**a**) Plain image of a baboon; (

**b**) cipher image by the correct secret key; (

**c**) cipher image by the changed secret key; (

**d**) information variation between two cipher images.

**Figure 6.**Key sensitivity in the decryption part. (

**a**) Plain image of a baboon; (

**b**) decrypted image with correct secret key; (

**c**) decrypted image with wrong secret key; (

**d**) difference between two decrypted images.

**Figure 7.**(

**a**) Histogram of a plain image of a baboon. (

**b**) Histogram of a plain image of a couple. (

**c**) Histogram of a plain image of a bridge. (

**d**) Histogram of a cipher image of a baboon. (

**e**) Histogram of a cipher image of a couple. (

**f**) Histogram of a cipher image of a bridge.

**Figure 8.**Correlation of adjacent pixel pairs for the baboon image. (

**a**) Plain image by horizontal direction; (

**b**) plain image by vertical direction; (

**c**) plain image by diagonal direction; (

**d**) cipher image by horizontal direction; (

**e**) cipher image by vertical direction; (

**f**) cipher image by diagonal direction.

**Figure 9.**The cipher image of a baboon transmitted in the channel containing data loss of (

**a**) $6.25\%$, (

**b**) $25\%$, or (

**c**) $50\%$ and the corresponding decrypted image of the baboon containing data loss of (

**d**) $6.25\%$, (

**e**) $25\%$, or (

**f**) $50\%$.

**Figure 10.**The decrypted image of a baboon transmitted by the channel containing AWGN with zero-mean and a standard deviation of (

**a**) 40, (

**b**) 60, or (

**c**) 80.

**Figure 11.**PSNR of decrypted images by the proposed scheme or [7] in different situations. AWGN with zero-mean and with a standard deviation of 40, 60, or 80 in the channel for the images: (

**a**) baboon, (

**b**) couple, (

**c**) bridge, and (

**d**) Lena; the data loss is $1/16$, $1/4$, or $1/2$ in the channel for: (

**e**) baboon, (

**f**) couple, (

**g**) bridge, and (

**h**) Lena.

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

Baboon/Cipher Baboon | 0.7596/0.0013 | 0.8190/0.0006 | 0.7056/0.0042 |

Couple/Cipher Couple | 0.9559/0.0057 | 0.9359/0.0075 | 0.9056/0.0043 |

Bridge/Cipher Bridge | 0.8861/0.0073 | 0.9079/0.0129 | 0.8416/0.0072 |

Lena/Cipher Lena | 0.9705/0.0089 | 0.9426/0.0125 | 0.9178/0.0006 |

Pepper/Cipher Pepper | 0.9603/0.0024 | 0.9540/0.0309 | 0.9217/0.0059 |

Sailboat/Cipher Sailboat | 0.9319/0.0050 | 0.9368/0.0353 | 0.8952/0.0025 |

Average of Plane/Cipher | 0.9107/0.0050 | 0.9160/0.0166 | 0.8646/0.0041 |

**Table 2.**Correlation coefficients of adjacent pixel pairs in the encrypted Lena image for different cryptosystems.

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

Proposed scheme | 0.0089 | 0.0125 | 0.0006 |

[38] | 0.0006 | 0.0013 | 0.0019 |

[39] | 0.0190 | 0.0127 | 0.0012 |

[40] | 0.0015 | 0.0002 | 0.0040 |

[41] | 0.0054 | 0.0045 | 0.0031 |

NPCR (%) | UACI (%) | |
---|---|---|

Baboon | 98.25 | 19.51 |

Couple | 99.22 | 31.95 |

Bridge | 98.93 | 23.34 |

Lena | 98.84 | 24.03 |

Pepper | 98.83 | 23.23 |

Sailboat | 99.50 | 33.31 |

Average | 98.93 | 25.90 |

Test Name | p-Value | Conclusion | |
---|---|---|---|

Approximate Entropy | 0.643855 | Pass | |

Block Frequency | 0.559242 | Pass | |

Cumulative Sums (Forward) | 0.889834 | Pass | |

Cumulative Sums (Reverse) | 0.867819 | Pass | |

FFT | 0.229310 | Pass | |

Frequency | 0.977662 | Pass | |

Linear Complexity | 0.625158 | Pass | |

Longest Runs of Ones | 0.997956 | Pass | |

Nonperiodic Templates | 0.983870 | Pass | |

Overlapping Template of All Ones | 0.430142 | Pass | |

Random Excursions | $x=-4$ | 0.418539 | Pass |

$x=-3$ | 0.317595 | Pass | |

$x=-2$ | 0.722055 | Pass | |

$x=-1$ | 0.107461 | Pass | |

$x=1$ | 0.285961 | Pass | |

$x=2$ | 0.840700 | Pass | |

$x=3$ | 0.291290 | Pass | |

$x=4$ | 0.167648 | Pass | |

Random Excursions Variant | $x=-9$ | 0.430201 | Pass |

$x=-8$ | 0.320963 | Pass | |

$x=-7$ | 0.189480 | Pass | |

$x=-6$ | 0.138033 | Pass | |

$x=-5$ | 0.121120 | Pass | |

$x=-4$ | 0.183213 | Pass | |

$x=-3$ | 0.456083 | Pass | |

$x=-2$ | 0.709549 | Pass | |

$x=-1$ | 0.333201 | Pass | |

$x=1$ | 0.258912 | Pass | |

$x=2$ | 0.232093 | Pass | |

$x=3$ | 0.145987 | Pass | |

$x=4$ | 0.131251 | Pass | |

$x=5$ | 0.149733 | Pass | |

$x=6$ | 0.165778 | Pass | |

$x=7$ | 0.469584 | Pass | |

$x=8$ | 0.514143 | Pass | |

$x=9$ | 0.449499 | Pass | |

Rank | 0.549642 | Pass | |

Runs | 0.730847 | Pass | |

Serial | p-value${}_{1}$ | 0.785632 | Pass |

p-value${}_{2}$ | 0.711362 | Pass | |

Universal Statistic | 0.863189 | Pass |

Encryption (s) | Decryption (s) | |
---|---|---|

Baboon | 0.17 | 1.5 |

Couple | 0.16 | 1.5 |

Bridge | 0.17 | 1.5 |

Lena | 0.17 | 1.6 |

Mean | 0.17 | 1.5 |

Plain Image | Cipher Image | |
---|---|---|

Baboon | 7.1352 | 7.9966 |

Couple | 6.3990 | 7.9967 |

Bridge | 7.7282 | 7.9966 |

Lena | 7.4962 | 7.9959 |

Mean | 7.1897 | 7.9965 |

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## Share and Cite

**MDPI and ACS Style**

Shen, Q.; Liu, W.; Lin, Y.; Zhu, Y.
Designing an Image Encryption Scheme Based on Compressive Sensing and Non-Uniform Quantization for Wireless Visual Sensor Networks. *Sensors* **2019**, *19*, 3081.
https://doi.org/10.3390/s19143081

**AMA Style**

Shen Q, Liu W, Lin Y, Zhu Y.
Designing an Image Encryption Scheme Based on Compressive Sensing and Non-Uniform Quantization for Wireless Visual Sensor Networks. *Sensors*. 2019; 19(14):3081.
https://doi.org/10.3390/s19143081

**Chicago/Turabian Style**

Shen, Qian, Wenbo Liu, Yi Lin, and Yongjun Zhu.
2019. "Designing an Image Encryption Scheme Based on Compressive Sensing and Non-Uniform Quantization for Wireless Visual Sensor Networks" *Sensors* 19, no. 14: 3081.
https://doi.org/10.3390/s19143081