## 4 The SEME-TDHM Algorithm
### Encryption Process
Full-channel random inverse scrambling: Reorder the rows of the image based on the index values generated by the chaotic sequence and cyclically shift the index sequence in an orderly manner to accomplish the inverse disordering of the rows. Columns are the same way. The process involving pseudo-random code is as Algorithm 3;
```
Input: recovered block matrices \(\boldsymbol{B1}\), \(\boldsymbol{B2}\), \(\boldsymbol{B3}\), sequence \(\boldsymbol{s2}\) of index \(\boldsymbol{1-(3\times N)}\), sequence \(\boldsymbol{s1}\) of index \(\boldsymbol{1-M}\), random sequences \(\boldsymbol{uu1}\), \(\boldsymbol{uu2}\) \(\boldsymbol{P1}\) = [B1, B2, B3]; for I = 2:3 \(\times\) N  sd = circshift(s2,uu2(i - 1));  temp_d = \(\boldsymbol{P1}\)(;i); \(\boldsymbol{P1}\)(s4,i) = temp_d; end  temp_d = \(\boldsymbol{P1}\)(;.1); \(\boldsymbol{P1}\)(s2,1) = temp_d; for i = 2:M  s3 = circshift(s1,uu1(i - 1));  temp_c = \(\boldsymbol{P1}\)(i,:); \(\boldsymbol{P1}\)(i, s3) = temp_c; end  temp_c = \(\boldsymbol{P1}\)(1,:); \(\boldsymbol{P1}\)(1, s1) = temp_c;
```

**Algorithm 3** Full-channel random inverse scramble

Step5. Single-channel random inverse scrambling: After dividing the image matrix P1, apply cyclic shifting to the block pixel matrix following the division. This movement is in the opposite direction of the encryption process. Three rounds of cycling are carried out as shown in the encryption process;

1. Inverse preprocessing: To obtain a decrypted image, recover pixel values in an orderly fashion through unfolding, transposing, shifting, and reconstructing. ## 5 Experimental Results and Performance Analysis

In this section, images of different sizes have been selected to verify the effectiveness of the proposed SEME-TDHM algorithm. The chosen experimental subjects are depicted in Figure 10, where figures (a), (b), and (c) are the original images, encrypted, and decrypted images, respectively. As observed in the figures, the encrypted images do not have any information about the original image, and the decrypted image does not contain invalid noise. The employed SEME-TDHM algorithm effectively encrypts and decrypts images, yielding commendable results. ### Key Space and Key Sensitivity Analysis

One of the performance criteria for image encryption algorithms is their ability to withstand brute-force attacks effectively, which necessitates a sufficiently large key space. In this paper, the SEME-TDHM algorithm's secret key primarily comprises the parameters \(a_{0}\), \(b_{0}\) of the hyperchaotic map, two discard lengths, and the initial values \(x_{0}\), \(y_{0}\) of the hyperchaotic map. To ensure the security and effectiveness of the encryption, the key space must be no less than \(2^{128}\)[32; 33]. Given the computational precision of \(10^{-15}\), the SEME-TDHM algorithm's key space is \(10^{150}\), which exceeds \(2^{256}\), significantly surpassing the minimum required key space. Consequently, the key space is sufficiently large to provide protection against brute-force attacks and potential threats from other cryptanalysis algorithms. Meanwhile, the encrypted Candy images were decrypted using four keys slightly different from the correct key, and the decrypted results are shown in Figure 11. It is clear that these images are completely different from the correctly decrypted images. Therefore, it can be concluded that the SEME-TDHM algorithm has the ability to resist known plaintext attacks. Figure 10: Encrypted and decrypted images, original images are (**al**) Candy, \(256\times 256\), (**a**2) Baboon, \(512\times 512\), (**a3**) House, \(614\times 409\), (**a4**) Bird, \(768\times 512\), (**a5**) Pepper, \(1000\times 1000\), (**a6**) Aircraft, \(1024\times 1024\); its corresponding encrypted images are (**b1–b6**); its corresponding decrypted images are (**c1–c6**). ### Information Entropy

Information entropy is a measure of the uncertainty of the average of a random variable. It evaluates the degree of diversity and unpredictability present in image data.