A. Blocks-permutation

The plain image is each of the two multiplex images obtained in Section 3. The plain image is decomposed into small blocks of the same size; let us choose blocks size of (4 � 4) pixels. In fact, increasing the number of blocks by using smaller block size resulted in a lower correlation and higher entropy; then, the intelligible information contained in the image will be reduced.

The permutation of blocks is realized as follows:


*Multiple-Image Fusion Encryption (MIFE) Using Discrete Cosine Transformation (DCT)… DOI: http://dx.doi.org/10.5772/intechopen.92369*

Number all the blocks of the plain image obtained in step 1, and adjust their positions with the previous permutation of step 3. Then, the image obtained is a block image permuted.

The values *x*01, *r*01, *x*02, *r*<sup>02</sup> are calculated through Eqs. (9) and (10). In this process, we subdivide each multiplex image *I*i, (*i = 1, 2*) in two parts, *P*<sup>1</sup> and *P*<sup>2</sup> of same size.

$$x\_{0i} = (x\_0 + mean(I\_i) / 2\text{55}) \text{mod1} \tag{9}$$

$$r\_{0i} = \left(r\_0 + \mathbf{0}.\mathbf{1} \times \max\left(\mathbf{S}\_1, \mathbf{S}\_2\right) / \mathbf{N} \times \mathbf{M} \times \mathbf{2}^9\right.\tag{10}$$

where*, S*<sup>1</sup> is the sum of pixels' intensities of the first part *P*<sup>1</sup> of the multiplex image *I*i, and *S*<sup>2</sup> for *P*2**.** *x*<sup>0</sup> ϵ [0, 0.9], *r* ϵ [0, 4.9].

#### B. Diffusion of the scrambled images

is performed. A simple rotation is performed on each of these blocks before spectral multiplexing, to prevent from information overlap. **Figure 4** illustrates the description of the process. It is possible to multiplex a large number of target images by selecting a smaller size of the filter. However, in this case, the recovered images will be highly altered. To keep a good quality of reconstructed images while maintaining a large number of target images to encrypt, we chose to group these images in two

This section presents the proposed cryptosystem, which comprises blockspermutation and diffusion steps using chaotic generators. **Figure 5** illustrates the

The plain image is each of the two multiplex images obtained in Section 3. The plain image is decomposed into small blocks of the same size; let us choose blocks size of (4 � 4) pixels. In fact, increasing the number of blocks by using smaller block size resulted in a lower correlation and higher entropy; then, the intelligible

1.Divide the plain image *I* I of size *M* � *M* into *k* blocks of size (4 � 4), with

3.Repeat step 2 to generate a new sequence, using new initial condition and control parameters *x*<sup>02</sup> and *r*02. This second sequence is to permute the small

4. Sort the chaotic sequence P in ascending order, and get a new sequence *P*<sup>0</sup>

. Therefore, the sequence *x*01, *r*01, *x*02, *r*<sup>02</sup> is the

¼

sequence *X* obtained are ranged in a row vector *P* of size 1, ð Þ*k* .

2.Use initial condition and control parameters *x*01, *r*<sup>01</sup> of Logistic-May system to generate a chaotic sequence by iterating *k* times Eq. (5). The values of the

multiplex images of the same size.

*Multimedia Information Retrieval*

entire process.

**Figure 5.** *Encryption scheme.*

> *<sup>k</sup>* <sup>¼</sup> *<sup>M</sup>* <sup>4</sup> � *<sup>M</sup>* 4

*P*0 *t* 

**28**

*<sup>k</sup>* ¼ *P*<sup>0</sup>

*<sup>t</sup>*1; *P*<sup>0</sup>

*2.5.1 Encryption process*

A. Blocks-permutation

**2.5 Proposed encryption/decryption scheme**

information contained in the image will be reduced. The permutation of blocks is realized as follows:

blocks of the second multiplex image.

*<sup>t</sup>*2, … *P*<sup>0</sup> *tk*

permutation of the sequence 1, 2, … , *k.*

At this level, the two scrambled images are combined in order to create the final hybrid encrypted images that would be difficult to crack. The May-Gaussian and Gaussian-Gompertz systems in Eqs. (6) and (7) are used as pseudo random generators to generate two chaotic sequences after 2 M � 2 M iterations. These values are arranged in two arrays *W* and *T* of sizes 2 M � 2 M, respectively, where M represents the number of rows and columns of each scrambled image. W and *T* are converted into real values in unit 8 format; *(W = uint8(W*�*255); T = uint8(T*�*255)).* The initial conditions and control parameters of the two pseudo random numbers generators are *x*p1**,** *r*p1 and *x*p2, *r*p2, α, respectively, for May-Gaussian and Gaussian-Gompertz systems. These parameters are determined with Eqs. (11) and (12).

$$
\pi\_{p^i} = \pi\_0 + \mathbf{0.1} \times \min\left(I\_i\right) / 2\mathbf{56} \tag{11}
$$

$$r\_{pi} = r + \mathbf{0.1} \times \min\left(I\_i + \mathbf{1}\right) / \max\left(I\_i + \mathbf{2}\right) \tag{12}$$

where *max (Ii)* and *min (Ii)* are, respectively, maximum and minimum pixel's intensities values of *I*i. *x*<sup>0</sup> ϵ [0, 0.9], *r* ϵ (0, 4.9].

$$\mathbf{W} = \begin{pmatrix} \frac{\mathbf{w}\_{11}}{\mathbf{w}\_{21}} \Big|\_{\mathbf{w}\_{22}}^{\mathbf{w}\_{12}} \\ \mathbf{w}\_{21} \Big|\_{\mathbf{w}\_{22}} \end{pmatrix}; \ \mathbf{T} = \begin{pmatrix} \frac{\mathbf{t}\_{11} \| \mathbf{t}\_{12}}{\mathbf{t}\_{21} \| \mathbf{t}\_{22}} \end{pmatrix} \tag{13}$$

The arrays *W* and *T* are divided into four sub-blocks of same size M � M. The two scrambled images *I1* and *I2* are linearly combined with the sub-blocks of *W* and *T* using the following equations:

$$C\_1(i,j) = \left[w\_{11} \times I\_1(i,j) + w\_{12} \times I\_2(i,j)\right] \text{mod}256 \oplus floor(t\_{11} \times t\_{21}) \times 10^{15} \text{()}\tag{14}$$

$$\mathbf{C\_2(i,j)} = \left[ w\_{21} \times I\_1(i,j) + w\_{22} \times I\_2(i,j) \right] \mathbf{mod256} \oplus \mathbf{floor}(t\_{12} \times t\_{22}) \times \mathbf{10^{15}} \mathbf{)} \mathbf{j} \tag{15}$$

where *C*1ð Þ *i*, *j* and *C*2ð Þ *i*, *j* are the two encrypted hybrid images of the cryptosystem, and ⊕ is the bit wise XOR operator. The mixed product *tij* � *tji* in the above relations enhances the quality of the merged images.

#### *2.5.2 Decryption process*

In the decryption process, the encrypted images are first decomposed using Cramer's rule in order to recover the scrambled images. Knowing the fusion keys (*x*p1, *r*p1, *x*p2, *r*p2, α), the receiver can get the images *I1* and *I2* by solving the system of equations below:

$$\begin{cases} \left(I\_1[i,j] \times w\_{11} + I\_2[i,j] \times w\_{12}\right)\_{md256} = C\_1 \left(floor(t\_{11} \times t\_{21}) \times \mathbf{10^{15}}\right) \\\ \left(I\_1[i,j] \times w\_{21} + I\_2[i,j] \times w\_{22}\right)\_{md256} = C\_2 \left(floor(t\_{12} \times t\_{22}) \times \mathbf{10^{15}}\right) \end{cases} \tag{16}$$

Then, the two multiplex images can be obtained easily by decrypting *I1* and *I2* through reverse permutation operations.
