# Image Encryption Scheme Based on Mixed Chaotic Bernoulli Measurement Matrix Block Compressive Sensing

^{*}

## Abstract

**:**

## 1. Introduction

- A hybrid chaotic sequence measurement matrix satisfying Bernoulli’s property based on Chebyshev map and a logistic map, which is easy to implement in hardware, is designed, and the performance of the resulting measurement matrix is analyzed and applied to BCS.
- The SHA-256 value of the plaintext image is used to calculate the initial value of the chaotic mapping for the encryption algorithm part, which increases the relevance of the algorithm to the plaintext image and improves the ability of the scheme to resist the selective plaintext attack.
- A “no repetition” scrambling algorithm and a Galois domain-based two-way diffusion algorithm are proposed to encrypt the measurement value matrix and improve the security of the BCS framework.
- In the reconstruction stage, the SPL reconstruction algorithm, based on DDWT, is used, which, combined with the hybrid chaotic sequence measurement matrix proposed in this paper, can achieve a much higher reconstruction quality of decrypted images with low compression ratio.

## 2. Preliminaries

#### 2.1. Block Compressive Sensing

#### 2.2. Measurement Matrix Generation Algorithm

#### 2.2.1. Hybrid Chebyshev–Logistic Bernoulli Sequence

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

#### 2.2.2. Construction of a Measurement Matrix Based on MCLB Sequence

- Set the parameters $\omega =4$, $\lambda =2$ of Equations (4) and (6), and given the initial values ${x}_{0}$, ${y}_{0}$, generate Chebyshev chaotic sequences $\left\{{x}_{n}\right\}$ and logistic chaotic sequences $\left\{{y}_{n}\right\}$ of the desired length iteratively from Equations (4) and (6).
- Binary quantization of the chaotic sequences $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ using Equations (9) and (10), respectively, to obtain the chaotic spread spectrum sequence $\left\{{x}_{n}^{\prime}\right\}$ and $\left\{{y}_{n}^{\prime}\right\}$, and XOR operation to generate a hybrid chaotic sequence $\left\{{z}_{n}\right\}$.
- The mixed chaotic sequence is cyclically shifted by columns to generate a measurement matrix of desired size, which is called the mixed Chebyshev and logistic Bernoulli matrix (MCLBM). Finally, the MCLBM is orthogonalized and normalized to obtain the final measurement matrix.

#### 2.2.3. Simulation Test Based on MCLBM

## 3. The Proposed Algorithm

#### 3.1. Block Compression Sampling and Quantification Process

- Generate the MCLB sequence $\left\{{z}_{j},j=1,2,\cdots ,L\right\}$, from the steps in Section 2.2.2, and $L=m\times n$, $n=B\times B$, $m=\mathrm{floor}(CR\times n)$.
- Generate a measurement matrix ${\Phi}_{0}$ of size $m\times n$ by cyclically shifting $\left\{{z}_{j}\right\}$ by columns:$${\Phi}_{0}=\left(\begin{array}{cccc}{z}_{1}& {z}_{m+1}& \cdots & {z}_{mn-m+1}\\ {z}_{2}& {z}_{m+2}& \cdots & {z}_{mn-m+2}\\ \vdots & \vdots & \ddots & \vdots \\ {z}_{m}& {z}_{2m}& \cdots & {z}_{mn}\end{array}\right)$$
- Express ${\Phi}_{0}$ as a row vector as ${\Phi}_{0}=\left({\alpha}_{1}^{T},{\alpha}_{2}^{T},\cdots ,{\alpha}_{M}^{T}\right)$. Let ${\beta}_{1}={\alpha}_{1}$ and use Equations (16) and (17) to normalize the row vectors.$${\beta}_{r}={\alpha}_{r}-{\displaystyle \sum _{i=1}^{r-1}\frac{\langle {\beta}_{i},{\alpha}_{r}\rangle}{\langle {\beta}_{i},{\beta}_{i}\rangle}}{\beta}_{i}\text{\hspace{1em}}r=2,3,\cdots ,M$$$${\eta}_{j}=\frac{{\beta}_{j}}{\Vert {\beta}_{j}\Vert}j=1,2,\cdots ,M$$
- Compress the plaintext image P after blocked and obtain the measurement value ${C}_{i}$ of each block.$${C}_{i}={\Phi}_{i}{S}_{i}i=1,2,\cdots ,{n}_{B}$$

#### 3.2. Cipher Sequence Generation Algorithm

#### 3.3. Image Encryption Algorithm

- First, keep only one of the recurring pseudo-random numbers in the pseudo-random sequence $X$ (i.e., the first occurrence), then add the values in $\left\{1,2,\cdots ,m{n}_{B}\right\}$ that do not appear in $X$ to the end of $X$ in descending order, and note ${X}_{1}$. ${X}_{1}$ is then used to displace $C$ without repetition. $C$ is then expanded column-wise into a one-dimensional vector, denoted ${C}_{1}$. Finally, ${C}_{1}\left({X}_{1}\left(i\right)\right)$ is swapped with ${C}_{1}\left({X}_{1}\left(m{n}_{B}-i+1\right)\right)$ to give the disordered matrix ${C}_{2}$.
- Select the first $m{n}_{B}$ values of the sequence $Y$ to form the sequence ${Y}_{1}$. ${C}_{3}$ is obtained by forward diffusion of ${C}_{2}$ using ${Y}_{1}$ as follows.$$\{\begin{array}{lll}{C}_{3}\left(1\right)={C}_{3}\left(0\right)\times {Y}_{1}\left(1\right)\times {C}_{2}\left(1\right)& & \\ {C}_{3}\left(i\right)={C}_{3}\left(i-1\right)\times {Y}_{1}\left(i\right)\times {C}_{2}\left(i\right)& & i=2,3,\cdots ,m{n}_{B}\end{array}$$
- Select the values from $m{n}_{B}+1$ to $2m{n}_{B}$ of the sequence $Y$ to form the sequence ${Y}_{2}$. Use ${Y}_{2}$ to perform backward diffusion on ${C}_{3}$ to obtain ${C}_{4}$, as follows.$$\{\begin{array}{lll}{C}_{4}\left(m{n}_{B}\right)={C}_{4}\left(0\right)\times {Y}_{2}\left(m{n}_{B}\right)\times {C}_{3}\left(m{n}_{B}\right)& & \\ {C}_{4}\left(i\right)={C}_{4}\left(i+1\right)\times {Y}_{2}\left(i\right)\times {C}_{3}\left(i\right)& & i=m{n}_{B}-1,m{n}_{B}-2,\cdots ,1\end{array}$$

#### 3.4. Image Decryption and Reconstruction Process

- Use the key K to generate the chaotic mapping initial values ${x}_{0}^{\prime}$, ${\omega}^{\prime}$, ${y}_{0}^{\prime}$, and ${\lambda}^{\prime}$, and generate $X$ and $Y$ by Equation (23).
- The received ciphertext image ${C}_{5}$ is sequentially subjected to backward diffusion, forward diffusion, and scrambling algorithms to obtain the quantized measurement value matrix ${P}_{0}$.
- The measurement matrix $C$ is recovered by inverse quantization of the quantized measurement matrix ${P}_{0}$ by the following equation:$$C=\frac{{P}_{0}\times \left(\mathrm{max}-\mathrm{min}\right)}{255}+\mathrm{min}$$
- The parameters $\omega ,\text{}\lambda $ and the initial value ${x}_{0},\text{}{y}_{0}$ of the MCLB transmitted by the sender are used to iterate and generate the MCLBM, that is, the measurement matrix ${\Phi}_{i}$.
- The DDWT-SPL algorithm is used to reconstruct the recovered measurement matrix $C$ to obtain a reconstructed original image $P$ of size $M\times N$.

## 4. Simulation and Performance Analysis

#### 4.1. Image Encryption Algorithm

#### 4.2. Impact of Important Parameters on Encryption and Decryption Performance

## 5. Statistical Analysis

#### 5.1. Key Space

#### 5.2. Key Sensitivity

_{0}, K

_{1}, and K

_{2}with bit modifications, which are shown below.

_{0}= fb1197ae40dd90f096474ebf6a25494802027aa0a8b8f0f81c9d9a2e623c0b0a

_{1}= eb1197ae40dd90f006474ebf6a25494802027aa0a8b8f0f81c9d9a2e623c0b0a

_{2}= eb1197ae40dd90f006474ebf6a25494802027aa0a8b8f0f81c9e9a2e623c0b0a

_{0}, K

_{1}, and K

_{2}to obtain different cipher images, as shown in Figure 7b–e. Figure 7f–g shows the difference between two ciphertexts; as seen from Figure 7f–g, the “difference image” of two ciphertexts obtained by encrypting the same plaintext image shows a noise style, intuitively, it reflects that the two ciphertexts differ significantly; that is, the image encryption system has key sensitivity. Figure 7j–l shows the images decrypted by different “wrong” keys; it can be seen that the information of the original image cannot be obtained in the decrypted images, which indicates that the image decryption system is key sensitive.

#### 5.3. Histogram

#### 5.4. Information Entropy Analysis

#### 5.5. Correlation Analysis

#### 5.6. Choose Plaintext Attack

#### 5.7. Ciphertext Sensitivity

#### 5.8. Time Complexity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Reconstructed images with different measurement matrices at a compression ratio of 0.3. (

**a**) Original image; (

**b**) GM reconstructed image; (

**c**) BM reconstructed image; (

**d**) LBM reconstructed image; (

**e**) CBM reconstructed image; (

**f**) MCLBM reconstructed image.

**Figure 5.**Simulation results. (

**a1**–

**a4**) Plaintext images; (

**b1**–

**b4**) ciphertext images; (

**c1**–

**c4**) decrypted images.

**Figure 7.**Results of wrong key encryption and decryption. (

**a**) The plaintext image; (

**b**–

**e**) encrypted plaintext images using K, K

_{0}, K

_{1}, and K

_{2}, respectively; (

**f**) Difference image of encrypted image ciphertext using K and K

_{0}; (

**g**) Difference image of encrypted image ciphertext using K and K

_{1}; (

**h**) Difference image of encrypted image ciphertext using K and K

_{2}; (

**i**) Decrypt the (

**b**) ciphertext image using K; (

**j**–

**l**) Decrypt the (b) ciphertext image using K

_{0}, K

_{1}, and K

_{2}, respectively.

**Figure 8.**Histograms of the original and encrypted images. (

**a**) Plaintext histogram of Lena; (

**b**) plaintext histogram of Peppers; (

**c**) plaintext histogram of Cell; (

**d**) plaintext histogram of X-ray; (

**e**) ciphertext histogram of Lena; (

**f**) ciphertext histogram of Peppers; (

**g**) ciphertext histogram of Cell; (

**h**) ciphertext histograms of X-ray.

**Figure 9.**Distribution of adjacent pixels. (

**a**) Plaintext horizontal adjacent pixels; (

**b**) Plaintext vertical adjacent pixels; (

**c**) Plaintext diagonal adjacent pixels; (

**d**) Ciphertext horizontal adjacent pixels; (

**e**) Ciphertext vertical adjacent pixels; (

**f**) Ciphertext diagonal adjacent pixels.

Measurement Matrix | PSNR/dB | Time/s |
---|---|---|

GM | 32.82 | 3.50 |

BM | 32.25 | 3.70 |

LBM | 32.45 | 3.21 |

CBM | 32.57 | 3.56 |

MCLBM | 33.82 | 3.18 |

**Table 2.**Comparison of PSNR and consumption time for different block sizes with the same compression ratio.

Block Size | PSNR/dB | Time/s |
---|---|---|

32 × 32 | 32.7265 | 8.955823 |

64 × 64 | 32.9089 | 12.1209232 |

128 × 128 | 33.1612 | 153.747594 |

Image | CR | Ours | Reference [55] | Reference [56] |
---|---|---|---|---|

Lena | 0.25 | 32.7265 | 31.4240 | \ |

Peppers | 32.9463 | 30.6809 | \ | |

Lena | 0.5 | 36.7805 | 33.2299 | 23.3608 |

Peppers | 36.4142 | 32.1889 | 27.3366 | |

Lena | 0.75 | 41.0781 | 34.1313 | 34.7149 |

Peppers | 40.1133 | 33.1721 | 35.8463 |

Algorithm | Ours | Reference [55] | Reference [57] | Reference [58] |
---|---|---|---|---|

Key space | 2^{300} | 6.561 × 10^{87} | 10^{80} | 10^{75} |

Secret Keys | K and K_{0} | K and K_{1} | K and K_{2} |
---|---|---|---|

NPCR (%) | 99.6223 | 99.6074 | 99.6013 |

UACI (%) | 33.4190 | 33.4347 | 33.3847 |

Secret Keys | K and K_{0} | K and K_{1} | K and K_{2} |
---|---|---|---|

NPCR (%) | 99.5671 | 99.5725 | 99.6372 |

CR | Lena | Peppers | Cell | X-ray |
---|---|---|---|---|

0.25 | 7.9972 | 7.9971 | 7.9975 | 7.9975 |

0.5 | 7.9986 | 7.9988 | 7.9986 | 7.9983 |

0.75 | 7.9990 | 7.9990 | 7.9991 | 7.9989 |

Direction | Lenna | Peppers | Cell | X-ray | ||||
---|---|---|---|---|---|---|---|---|

Plaintext | Ciphertext | Plaintext | Ciphertext | Plaintext | Ciphertext | Plaintext | Ciphertext | |

Horizontal | 0.9845 | −0.0032 | 0.9751 | 0.0075 | 0.9893 | 0.0163 | 0.9985 | 0.0410 |

Vertical | 0.9746 | 0.0123 | 0.9763 | 0.0129 | 0.9904 | −0.0059 | 0.9976 | −0.0148 |

Diagonal | 0.9703 | −0.0071 | 0.9649 | −0.0051 | 0.9792 | 0.0215 | 0.9964 | −0.0082 |

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

Lena | 99.6246 | 33.5234 |

Peppers | 99.6109 | 33.4788 |

Cell | 99.6265 | 33.4929 |

X-ray | 99.6147 | 33.4934 |

Images | Lena | Peppers | ||
---|---|---|---|---|

Process | Encryption | Decryption | Encryption | Decryption |

CR = 0.25 | 1.007266 | 7.948557 | 0.903141 | 7.736676 |

CR = 0.5 | 1.344250 | 5.933346 | 1.190280 | 6.051189 |

CR = 0.75 | 1.640416 | 3.167031 | 1.956602 | 3.195703 |

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**MDPI and ACS Style**

Yang, C.; Pan, P.; Ding, Q.
Image Encryption Scheme Based on Mixed Chaotic Bernoulli Measurement Matrix Block Compressive Sensing. *Entropy* **2022**, *24*, 273.
https://doi.org/10.3390/e24020273

**AMA Style**

Yang C, Pan P, Ding Q.
Image Encryption Scheme Based on Mixed Chaotic Bernoulli Measurement Matrix Block Compressive Sensing. *Entropy*. 2022; 24(2):273.
https://doi.org/10.3390/e24020273

**Chicago/Turabian Style**

Yang, Chen, Ping Pan, and Qun Ding.
2022. "Image Encryption Scheme Based on Mixed Chaotic Bernoulli Measurement Matrix Block Compressive Sensing" *Entropy* 24, no. 2: 273.
https://doi.org/10.3390/e24020273