**2. Brief review on 1D chaotic systems used**

#### **2.1 1D logistic, May, Gaussian, and Gompertz maps**

The equations of 1D Logistic, May, Gaussian, and Gompertz maps are described from Eqs. (1) to (4), respectively [11].

*2.1.1 1D logistic map*

$$\mathbf{x}\_{n+1} = r\mathbf{x}\_n(\mathbf{1} - \mathbf{x}\_n) \tag{1}$$

*2.2.1 Logistic-May map (LM)*

*2.2.2 May-Gaussian (MG)*

*2.2.3 Gaussian-Gompertz*

around 2.5.

**25**

It is defined by Eq. (7)

*xn*þ<sup>1</sup> <sup>¼</sup> ð Þ *<sup>r</sup>=*<sup>5</sup> <sup>þ</sup> <sup>26</sup>

congruent for the three chaotic maps.

**2.3 Description of the watermarked process**

equal to 8.3.

Its equation is defined by Eq. (5)

*DOI: http://dx.doi.org/10.5772/intechopen.92369*

Eq. (6) defines the May-Gaussian (MG) map

*xn*þ<sup>1</sup> <sup>¼</sup> *xn* exp ðð Þ *<sup>r</sup>* <sup>þ</sup> <sup>10</sup> ð Þ <sup>1</sup> � *xn* Þ þ ð Þ *<sup>r</sup>* <sup>þ</sup> <sup>5</sup>

*xn*þ<sup>1</sup> ¼ ð Þ *xn* exp ðð Þ *r* þ 9 ð Þ 1 � *xn* Þ � ð Þ *r* þ 5 *xn*ð Þ 1 � *xn* mod2 (5)

mod2 (6)

 � ð Þ *<sup>r</sup>=*<sup>5</sup> <sup>þ</sup> <sup>26</sup> *xn* log *xn* mod2 (7)

where *xn* ∈½ � 0, 1 , *r*∈½ � 0, 5 , *α*∈½4*:*7, 17�. From its bifurcation diagram, the

*n*

where *xn* ∈½ � 0, 1 , *r*∈½ � 0, 5 , *α*∈½4*:*7, 17�. It has a mean Lyaponuv exponent

**Figure 2** illustrates the bifurcation diagram and the Lyaponuv exponent graphics of these maps. Referring to **Figure 2**, all the previous 1D chaotic systems present a wider chaotic range and a more uniform distribution of their density functions. Furthermore, the maximum Lyaponuv exponent values obtained are respectively 8.1, 5.6, and 2.5. Then, these combined 1D systems are more suitable for secure and high-speed encryption if the encryption algorithm is built around a good algebraic structure. Additively, in order to confirm the good performance of the previous pseudo random number generators, we performed the NIST statistical tests. Analysis of these results (see **Table 1**) showed that all the 15 tests were

Before multiplexing the target images, a binary information in the form of a logo was inserted in one of the target images. To do this, we used a simple watermarked algorithm, which makes the hidden message imperceptible in the watermarked image. Taking advantage of the benefits of DCT, it is possible to embed an information or watermark (text, logo, image) in low- or medium-frequency DCT coefficients. In fact, DCT decomposes an image into three frequency regions: low, medium, and high frequencies. It is recommended to insert the watermark in the

low- and medium-frequency regions of the host image in order to ensure imperceptibility [32]. In this work, we adopted the watermarking technique

Lyaponuv exponents are positive and belong to the range [2.5, 5.6].

<sup>4</sup> <sup>þ</sup> exp �*αx*<sup>2</sup>

<sup>4</sup> <sup>þ</sup> exp �*αx*<sup>2</sup>

*n*

where *xn* ∈ ½ � 0, 1 and *r*∈½ � 0, 5 . From its bifurcation diagram, we can observe that chaotic properties are excellent within [0, 5], with a maximum Lyaponuv exponent

*Multiple-Image Fusion Encryption (MIFE) Using Discrete Cosine Transformation (DCT)…*

where *xn* ∈½ � 0, 1 is the discrete state of the output chaotic sequence and *r* is the control parameter with values in the range (0, 4]. The chaotic behavior of the Logistic map is observed in the range [3.5, 4].

*2.1.2 May map*

$$\boldsymbol{\pi}\_{n+1} = \boldsymbol{\pi}\_n \exp\left(\boldsymbol{a}(\mathbf{1} - \boldsymbol{\pi}\_n)\right) \tag{2}$$

where *xn* ∈½ � 0, 10*:*9 and the control parameter *a* belongs to the range [0, 5].

*2.1.3 Gaussian map*

$$\mathbf{x}\_{n+1} = \exp\left(-a\mathbf{x}\_n^2\right) + c \tag{3}$$

where *α* ∈ [4.7, 17], *c* ∈ [�1, 1].

#### *2.1.4 Gompertz map*

$$
\mathbf{x}\_{n+1} = -b\mathbf{x}\_n \ln \mathbf{x}\_n \tag{4}
$$

where the control parameter *b* ∈ (0, *e*], *e* = 2.71829 … and is the exponential function.

#### **2.2 Combination of new 1D chaotic maps**

The chaotic properties of 1D Logistic, May, Gaussian, and Gompertz maps are not suitable to build a secure cryptosystem when they are used alone. To solve this problem, Zhou et al. [23] proposed to combine the different seed maps. **Figure 1** shows the new map obtained from a nonlinear combination of two different 1D chaotic maps.

**Figure 1.** *New chaotic scheme.*

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

#### *2.2.1 Logistic-May map (LM)*

**2. Brief review on 1D chaotic systems used**

from Eqs. (1) to (4), respectively [11].

*Multimedia Information Retrieval*

Logistic map is observed in the range [3.5, 4].

where *α* ∈ [4.7, 17], *c* ∈ [�1, 1].

**2.2 Combination of new 1D chaotic maps**

*2.1.1 1D logistic map*

*2.1.2 May map*

*2.1.3 Gaussian map*

*2.1.4 Gompertz map*

function.

chaotic maps.

**Figure 1.** *New chaotic scheme.*

**24**

**2.1 1D logistic, May, Gaussian, and Gompertz maps**

The equations of 1D Logistic, May, Gaussian, and Gompertz maps are described

where *xn* ∈½ � 0, 1 is the discrete state of the output chaotic sequence and *r* is the control parameter with values in the range (0, 4]. The chaotic behavior of the

where *xn* ∈½ � 0, 10*:*9 and the control parameter *a* belongs to the range [0, 5].

*xn*þ<sup>1</sup> <sup>¼</sup> exp �*αx*<sup>2</sup>

where the control parameter *b* ∈ (0, *e*], *e* = 2.71829 … and is the exponential

The chaotic properties of 1D Logistic, May, Gaussian, and Gompertz maps are not suitable to build a secure cryptosystem when they are used alone. To solve this problem, Zhou et al. [23] proposed to combine the different seed maps. **Figure 1** shows the new map obtained from a nonlinear combination of two different 1D

*n*

*xn*þ<sup>1</sup> ¼ *rxn*ð Þ 1 � *xn* (1)

*xn*þ<sup>1</sup> ¼ *xn* exp ð Þ *a*ð Þ 1 � *xn* (2)

<sup>þ</sup> *<sup>c</sup>* (3)

*xn*þ<sup>1</sup> ¼ �*bxn* ln *xn* (4)

Its equation is defined by Eq. (5)

$$\mathbf{x}\_{n+1} = (\mathbf{x}\_n \exp\left( (r+\mathfrak{H})(\mathbf{1}-\mathbf{x}\_n) \right) - (r+\mathfrak{H})\mathbf{x}\_n(\mathbf{1}-\mathbf{x}\_n))\mathbf{mod2} \tag{5}$$

where *xn* ∈ ½ � 0, 1 and *r*∈½ � 0, 5 . From its bifurcation diagram, we can observe that chaotic properties are excellent within [0, 5], with a maximum Lyaponuv exponent equal to 8.3.

*2.2.2 May-Gaussian (MG)*

Eq. (6) defines the May-Gaussian (MG) map

$$\mathbf{x}\_{n+1} = \left(\mathbf{x}\_n \exp\left( (r+1\mathbf{0})(\mathbf{1}-\mathbf{x}\_n) \right) + \frac{(r+5)}{4} + \exp\left( -\alpha \mathbf{x}\_n^2 \right) \right) \text{mod}\mathbf{2} \tag{6}$$

where *xn* ∈½ � 0, 1 , *r*∈½ � 0, 5 , *α*∈½4*:*7, 17�. From its bifurcation diagram, the Lyaponuv exponents are positive and belong to the range [2.5, 5.6].

#### *2.2.3 Gaussian-Gompertz*

It is defined by Eq. (7)

$$\mathbf{x}\_{n+1} = \left(\frac{(r/5+2\mathfrak{G})}{4} + \exp\left(-\alpha x\_n^2\right) - (r/5+2\mathfrak{G})\mathbf{x}\_n \log \mathbf{x}\_n\right) \text{mod}2\tag{7}$$

where *xn* ∈½ � 0, 1 , *r*∈½ � 0, 5 , *α*∈½4*:*7, 17�. It has a mean Lyaponuv exponent around 2.5.

**Figure 2** illustrates the bifurcation diagram and the Lyaponuv exponent graphics of these maps. Referring to **Figure 2**, all the previous 1D chaotic systems present a wider chaotic range and a more uniform distribution of their density functions. Furthermore, the maximum Lyaponuv exponent values obtained are respectively 8.1, 5.6, and 2.5. Then, these combined 1D systems are more suitable for secure and high-speed encryption if the encryption algorithm is built around a good algebraic structure. Additively, in order to confirm the good performance of the previous pseudo random number generators, we performed the NIST statistical tests. Analysis of these results (see **Table 1**) showed that all the 15 tests were congruent for the three chaotic maps.

#### **2.3 Description of the watermarked process**

Before multiplexing the target images, a binary information in the form of a logo was inserted in one of the target images. To do this, we used a simple watermarked algorithm, which makes the hidden message imperceptible in the watermarked image. Taking advantage of the benefits of DCT, it is possible to embed an information or watermark (text, logo, image) in low- or medium-frequency DCT coefficients. In fact, DCT decomposes an image into three frequency regions: low, medium, and high frequencies. It is recommended to insert the watermark in the low- and medium-frequency regions of the host image in order to ensure imperceptibility [32]. In this work, we adopted the watermarking technique

the host image; *Ds* represents the difference sum for all the pixels used in the block; and α is a constant threshold value selected. The value of α must be high to ensure the most hidden message imperceptibility in the watermarked image; α *є* [0, 255]. To illustrate an embedded process, as can be seen in **Figure 3**, we used a host image of size 512�512, and a binary watermarked image of size 64�64. We can notice from **Figure 3** that the binary image (watermarked) is recovered without loss

*Results of the watermarked process. (a) Host image (512* �512Þ*, (b) watermark (64* �64Þ*, (c) watermarked*

*Multiple-Image Fusion Encryption (MIFE) Using Discrete Cosine Transformation (DCT)…*

In order to protect the watermarked and host image from unauthorized access and noise attack, the watermarked image was encrypted with other images in a

In this section, *N* target images of size ð Þ *M*, *M* are combined into two images,

transformation (DCT) is first applied separately to each of the target images. In the second step, every spectrum is multiplied by a low-pass filter, of size (*M'*, *M'*) pixels, as indicated in **Figure 4**. In this manner, it is possible to reconstruct every target image through the relevant information contained in each block. At this step,

*C*<sup>p</sup> ¼ 1–ð Þ *size of multiplexed DCT spectral plane=size of N inputs images*

Then, after all of these target images are grouped together by a way of simple addition, the inverse discrete cosine transformation (IDCT) of the multiplex image

*<sup>=</sup><sup>N</sup>* � *<sup>M</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>*=<sup>N</sup>* (8)

each containing f g *N=*2 target images. As described in [24], discrete cosine

*cp* <sup>¼</sup> <sup>1</sup> � *<sup>M</sup>*<sup>2</sup>

of information.

*image, (d) recovered watermark.*

*DOI: http://dx.doi.org/10.5772/intechopen.92369*

**Figure 3.**

mixed process.

**Figure 4.**

**27**

*Spectral fusion of target images.*

**2.4 Spectral fusion of target images**

the compression rate *C*<sup>p</sup> is:

#### **Figure 2.**

*Bifurcation diagrams and Lyaponuv exponent graphics of combined chaotic maps, (a) and (d) logistic-may, (b) and (e) May-Gaussian, (c) and (f) Gaussian-Gompertz.*


#### **Table 1.**

*Statistical NIST tests results of 1,000,000 bits.*

described in [33] in which the message to hide is added to the medium-frequency region discrete cosine coefficients in selected pixel blocks of size 8 8. All the blocks satisfying the condition *Ds*> *Av α* are eligible blocks suitable for watermark embedding, where *Av* is the average for all pixels in the block considered from *Multiple-Image Fusion Encryption (MIFE) Using Discrete Cosine Transformation (DCT)… DOI: http://dx.doi.org/10.5772/intechopen.92369*

**Figure 3.**

*Results of the watermarked process. (a) Host image (512* �512Þ*, (b) watermark (64* �64Þ*, (c) watermarked image, (d) recovered watermark.*

the host image; *Ds* represents the difference sum for all the pixels used in the block; and α is a constant threshold value selected. The value of α must be high to ensure the most hidden message imperceptibility in the watermarked image; α *є* [0, 255].

To illustrate an embedded process, as can be seen in **Figure 3**, we used a host image of size 512�512, and a binary watermarked image of size 64�64. We can notice from **Figure 3** that the binary image (watermarked) is recovered without loss of information.

In order to protect the watermarked and host image from unauthorized access and noise attack, the watermarked image was encrypted with other images in a mixed process.

#### **2.4 Spectral fusion of target images**

In this section, *N* target images of size ð Þ *M*, *M* are combined into two images, each containing f g *N=*2 target images. As described in [24], discrete cosine transformation (DCT) is first applied separately to each of the target images. In the second step, every spectrum is multiplied by a low-pass filter, of size (*M'*, *M'*) pixels, as indicated in **Figure 4**. In this manner, it is possible to reconstruct every target image through the relevant information contained in each block. At this step, the compression rate *C*<sup>p</sup> is:

*C*<sup>p</sup> ¼ 1–ð Þ *size of multiplexed DCT spectral plane=size of N inputs images*

$$c\_p = \mathbf{1} - \left(\mathbf{M}^2/\mathbf{N} \times \mathbf{M}^2\right) = \mathbf{1} - \mathbf{1}/N \tag{8}$$

Then, after all of these target images are grouped together by a way of simple addition, the inverse discrete cosine transformation (IDCT) of the multiplex image

**Figure 4.** *Spectral fusion of target images.*

described in [33] in which the message to hide is added to the medium-frequency region discrete cosine coefficients in selected pixel blocks of size 8 8. All the blocks satisfying the condition *Ds*> *Av α* are eligible blocks suitable for watermark embedding, where *Av* is the average for all pixels in the block considered from

*Bifurcation diagrams and Lyaponuv exponent graphics of combined chaotic maps, (a) and (d) logistic-may,*

Frequency 0.98147 98/100 0.99680 100/100 0.99438 100/100 Block-frequency 0.6929 97/100 0.69842 98/100 0.678415 97/100 Cumulative-sums 0.78621 96/100 0.87124 97/100 0.9014 100/100 Runs 0.88052 99/100 0.92735 100/100 0.87246 98/100 Longest-runs 0.98654 99/100 0.99815 100/100 0.97729 98/100 Rank 0.54702 97/100 0.57914 98/100 0.5873 99/100 FFT 0.87531 97/100 0.89678 98/100 0.82670 98/100 Nonoverlapping-templates 0.78951 100/100 0.75091 99/100 0.77856 98/100 Overlapping-templates 0.28435 99/100 0.18942 97/100 0.25167 98/100 Universal 0.38277 99/100 0.34834 98/100 0.37051 100/100 Approximate entropy 0.45393 98/100 0.49357 99/100 0.41560 98/100 Random-excursions 0.195257 60/60 0.192410 59/60 0.19478 59/60 Random-excursions Variant 0.14358 58/60 0.13871 57/60 0.15120 59/60 Serial 0.42962 97/100 0.47359 99/100 0.41757 97/100 Linear-complexity 0.08945 98/100 0.32876 100/100 0.15762 98/100 Final result success success success

**May-Gaussian map (MG)**

**p-Value Result p-Value Result p-Value Result**

**Gaussian-Gompertz map**

**(LM)**

*(b) and (e) May-Gaussian, (c) and (f) Gaussian-Gompertz.*

*Multimedia Information Retrieval*

**Statistical test Logistic-May map**

**Figure 2.**

**Table 1.**

**26**

*Statistical NIST tests results of 1,000,000 bits.*

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

*Multiple-Image Fusion Encryption (MIFE) Using Discrete Cosine Transformation (DCT)…*

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

where*, S*<sup>1</sup> is the sum of pixels' intensities of the first part *P*<sup>1</sup> of the multiplex

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

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).

where *max (Ii)* and *min (Ii)* are, respectively, maximum and minimum pixel's

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

*<sup>C</sup>*1ð Þ¼ *<sup>i</sup>*, *<sup>j</sup>* ½ Þ *<sup>w</sup>*<sup>11</sup> � *<sup>I</sup>*1ð Þþ *<sup>i</sup>*, *<sup>j</sup> <sup>w</sup>*<sup>12</sup> � *<sup>I</sup>*2ð Þ *<sup>i</sup>*, *<sup>j</sup>* mod256 <sup>⊕</sup> *floor t*ð Þ� <sup>11</sup> � *<sup>t</sup>*<sup>21</sup> <sup>10</sup><sup>15</sup>Þ� (14) *<sup>C</sup>*2ð Þ¼ *<sup>i</sup>*, *<sup>j</sup>* ½ Þ *<sup>w</sup>*<sup>21</sup> � *<sup>I</sup>*1ð Þþ *<sup>i</sup>*, *<sup>j</sup> <sup>w</sup>*<sup>22</sup> � *<sup>I</sup>*2ð Þ *<sup>i</sup>*, *<sup>j</sup>* mod256 <sup>⊕</sup> *floor t*ð Þ� <sup>12</sup> � *<sup>t</sup>*<sup>22</sup> <sup>10</sup><sup>15</sup>Þ� (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

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

arranged in two arrays *W* and *T* of sizes 2 M � 2 M, respectively, where M

*x*0*<sup>i</sup>* ¼ ð Þ *x*<sup>0</sup> þ *mean I*ð Þ*<sup>i</sup> =*255 mod1 (9)

*xpi* ¼ *x*<sup>0</sup> þ 0*:*1 � min ð Þ *Ii =*256 (11)

ð13Þ

*rpi* ¼ *r* þ 0*:*1 � min ð Þ *Ii* þ 1 *=* max ð Þ *Ii* þ 2 (12)

*<sup>r</sup>*0*<sup>i</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>0</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>1</sup> � max ð Þ *<sup>S</sup>*1, *<sup>S</sup>*<sup>2</sup> *<sup>=</sup><sup>N</sup>* � *<sup>M</sup>* � 29 (10)

block image permuted.

image *I*i, and *S*<sup>2</sup> for *P*2**.** *x*<sup>0</sup> ϵ [0, 0.9], *r* ϵ [0, 4.9].

intensities values of *I*i. *x*<sup>0</sup> ϵ [0, 0.9], *r* ϵ (0, 4.9].

relations enhances the quality of the merged images.

*T* using the following equations:

*2.5.2 Decryption process*

**29**

B. Diffusion of the scrambled images

*DOI: http://dx.doi.org/10.5772/intechopen.92369*

same size.

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

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 multiplex images of the same size.
