**5. Evaluation of the quality of the algorithm**

The quality assessment of the visual distortion of the fractal cover image was performed on the basis of the following metrics: normalized mean square error (NMSE) and peak signal-to-noise ratio (PSNR) (Eqs. (10) and (11)):

$$\text{NMSE} = \sum\_{\mathbf{x}, \mathbf{y}} \frac{\left(\mathbf{C}\_{\mathbf{x}, \mathbf{y}} - \mathbf{S}\_{\mathbf{x}, \mathbf{y}}\right)^2}{\sum\_{\mathbf{x}, \mathbf{y}} \left(\mathbf{C}\_{\mathbf{x}, \mathbf{y}}\right)^2} \tag{10}$$

**6. Analysis of the possibility of steganalysis by measuring the fractal**

*Fractal images with different starting point c. (а)* �*0.73949 + 0.16498\**i*; (b)* �*0.74549 + 0.37841\**i*; (c)*

The importance of steganographic watermark embedding is their steganographic attacks resistance. In this case, the problem arises of detecting the very fact of the introduction of the watermark into the cover image. For this purpose, the method of estimating the fractal dimension of the image before and after embedding the watermark can be used. Consider the problem of measuring the fractal dimension of color or black and white images after embedding the watermark into them [15]. As is known, a fractal is defined as a collection of objects for which the Hausdorff dimension is strictly greater than the topological dimension. The concept

A bounded fractal set in a Euclidean n-space will be self-similar if it is a union of different N (disjoint) reduced copies that can be scaled using a special scaling factor r. In accordance with the entered scaling factor, the fractal dimension D of set A can

> *<sup>D</sup>* <sup>¼</sup> log ð Þ *<sup>N</sup>* log 1ð Þ *=r*

where N is the total number of boxes L needed to cover the fractal set; 1/r is the

As a result, D is a dimension relative to the size of the box used to measure the

fractal image. It can be said that the fractal dimension is a measure of how

Let us consider the two most common methods for measuring dimension:

(12)

**dimension of the secret key before and after embedding**

of self-similarity is used to estimate the fractal dimension.

*Watermarks obtained during extraction using fractals shown in Figure 5.*

scaling factor of the box in relation to the image.

differential box-counting and triangulation method.

be obtained using Eq. (12).

**Figure 5.**

**Figure 6.**

�*0.80939 + 0.12388\**i*; (d)* �*0.63949 + 0.19098\**i*.*

*Using Algebraic Fractals in Steganography DOI: http://dx.doi.org/10.5772/intechopen.92018*

"complex" a self-similar figure is.

**93**

$$PSNR = XY \cdot \max\_{\mathbf{x}, \mathbf{y}} \frac{\left(\mathbf{C}\_{\mathbf{x}, \mathbf{y}}\right)^2}{\sum\_{\mathbf{x}, \mathbf{y}} \left(\mathbf{C}\_{\mathbf{x}, \mathbf{y}} - \mathbf{S}\_{\mathbf{x}, \mathbf{y}}\right)^2} \tag{11}$$

In the relations presented, *Cx*,*<sup>y</sup>* denotes a pixel with the coordinates (x, y) of the empty cover image, and *Sx*,*<sup>y</sup>* denotes the corresponding pixel of the filled image.

The calculation of the metrics was carried out with different sizes of the implemented secret key and digital watermark. The results are presented in **Table 1**.

To assess the secrecy of the developed system, a situation was simulated in which the attacker took possession of a secret key with an embedded watermark, and he also knows some parameters of the generated fractal, namely, the size of the rectangle l, the maximum number of iterations k and the image size m. The only unknown parameter is the starting point c. As an example, four fractals were generated with different values of the parameter c (**Figure 5**) and the watermark was extracted using the obtained images and the secret key containing the watermark presented in **Figure 4b**.

The results of the extraction of watermark are shown in **Figure 6**.

As can be seen from the figures, all the images obtained as a result of the experiment are so significantly different from the original OT. As a result, the extraction of information from the QR code becomes impossible. From the experiment, it is clear that to obtain the information it is necessary to know the parameters of the fractal key with high accuracy. The developed method makes it possible to reduce the probability of the substitution or theft of secret information in similar steganographic systems to almost zero.


**Table 1.** *Values of NMSE and PSNR with different sizes of embedded data.*

**Figure 5.**

**Figure 4** shows the results of extracting a digital watermark from the selected fractal. **Figure 4a** contains the original watermark and **Figure 4b** shows watermark

As a result of the algorithm, a watermark was extracted with high quality. Despite the fact that you can observe small pixel distortions in the extracted secret key, the received watermark as a QR code is completely identical to the original one.

The quality assessment of the visual distortion of the fractal cover image was performed on the basis of the following metrics: normalized mean square error

> *Cx*,*<sup>y</sup>* � *Sx*,*<sup>y</sup>* � �<sup>2</sup>

> > *<sup>x</sup>*,*<sup>y</sup> Cx*,*<sup>y</sup>*

*Cx*,*<sup>y</sup>* � �<sup>2</sup>

*<sup>x</sup>*,*<sup>y</sup> Cx*,*<sup>y</sup>* � *Sx*,*<sup>y</sup>*

� �<sup>2</sup> (10)

� �<sup>2</sup> (11)

*x*, *y*

P

P

In the relations presented, *Cx*,*<sup>y</sup>* denotes a pixel with the coordinates (x, y) of the empty cover image, and *Sx*,*<sup>y</sup>* denotes the corresponding pixel of the filled image. The calculation of the metrics was carried out with different sizes of the implemented secret key and digital watermark. The results are presented in **Table 1**. To assess the secrecy of the developed system, a situation was simulated in which the attacker took possession of a secret key with an embedded watermark, and he also knows some parameters of the generated fractal, namely, the size of the rectangle l, the maximum number of iterations k and the image size m. The only unknown parameter is the starting point c. As an example, four fractals were generated with different values of the parameter c (**Figure 5**) and the watermark was extracted using the obtained images and the secret key containing the water-

(NMSE) and peak signal-to-noise ratio (PSNR) (Eqs. (10) and (11)):

*NMSE* <sup>¼</sup> <sup>X</sup>

The results of the extraction of watermark are shown in **Figure 6**. As can be seen from the figures, all the images obtained as a result of the experiment are so significantly different from the original OT. As a result, the extraction of information from the QR code becomes impossible. From the experiment, it is clear that to obtain the information it is necessary to know the parameters of the fractal key with high accuracy. The developed method makes it possible to reduce the probability of the substitution or theft of secret information in similar

**Size of watermark, pixels Fractal key size, pixels NMSE PSNR** � 30 85 � 85 0.0021 31,1922 � 35 100 � 100 0.0031 29,4217 � 40 115 � 115 0.0056 26,7753 � 45 130 � 130 0.006 26,5334 � 50 150 � 150 0.0067 26,3538

*PSNR* ¼ *XY* � max *<sup>x</sup>*,*<sup>y</sup>*

extracted from the key.

*Fractal Analysis - Selected Examples*

mark presented in **Figure 4b**.

**Table 1.**

**92**

steganographic systems to almost zero.

*Values of NMSE and PSNR with different sizes of embedded data.*

**5. Evaluation of the quality of the algorithm**

*Fractal images with different starting point c. (а)* �*0.73949 + 0.16498\**i*; (b)* �*0.74549 + 0.37841\**i*; (c)* �*0.80939 + 0.12388\**i*; (d)* �*0.63949 + 0.19098\**i*.*

**Figure 6.** *Watermarks obtained during extraction using fractals shown in Figure 5.*
