**8. The measurement results of the fractal dimension of fractal keys**

To illustrate the possibility of detecting an embedded watermark in images, we consider the experimental results of the evaluation of the fractal dimension after embedding the watermark in a fractal image based on Julia set.

**Table 3** shows the results of measuring the fractal dimension of color fractal images in the form of a Julia set when embedding watermark by various known steganographic methods.

The results show that the steganographic introduction of data does not actually affect the value of the fractal dimension, which makes it impossible to illegally extract the watermark.


**9. Conclusions**

**Table 3.**

**Starting point** *c*

**Number of iterations**

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

> **Size of rectangle**

**Embedding method**

300 1.5 LSB 512 2.449 2.449

**Image size**

Koch G 2.172 2.165

Koch B 2.172 2.359

Block based 2.449 2.442 Koch R 2.449 2.427 Koch G 2.449 2.425 Koch B 2.449 2.450

Block based 2.541 2.546 Koch R 2.541 2.523 Koch G 2.541 2.523 Koch B 2.541 2.537

Block based 2.396 2.383 Koch R 2.396 2.474 Koch G 2.396 2.423 Koch B 2.396 2.487

Block based 2.183 2.183 Koch R 2.183 2.221 Koch G 2.183 2.323 Koch B 2.183 2.122

0.5 LSB 2.541 2.541

0.0005 LSB 2.396 2.396

**D of original fractal**

**D of fractal with watermark**

form of a Julia set.

**99**

escape time algorithm (Escape time algorithm).

watermark in the form of a QR code.

The secrecy of steganographic systems is based on the assumption that the attacker is not aware of the fact of the information being introduced. In the event that this fact becomes publicly available, it will not be difficult for an unauthorized user to extract secret data or delete a given watermark. To solve this problem, it was proposed to use an additional container in the form of an algebraic fractal in the

*Changing the value of the fractal dimension of images during steganographic embedding.*

3000 LSB 2.183 2.183

Fractal generation is carried out using predefined secret parameters using the

Embedding a digital watermark in a container was carried out in two stages. At the first stage, the watermark is added to the generated fractal. The resulting image is embedded in the original container in JPEG format. As a result, the original image has almost no visual distortion. The measurement of the values of the NMSE and PSNR metrics confirmed the high level of embedding quality and extraction of the

To confirm the high level of secrecy, an experiment was conducted, during which an attempt was made to replace the original secret key. The experiment


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

**Table 3.**

The numerical data of the absolute errors of measurement of the dimension with the help of triangulation and DBC methods show that they give comparable results. The error of calculations for both methods does not exceed 5%. For practical use, it is advisable to use both methods, and the dimension values to find by averaging the results of both methods. In this case, the total error of calculations

**8. The measurement results of the fractal dimension of fractal keys**

embedding the watermark in a fractal image based on Julia set.

**Size of rectangle**

To illustrate the possibility of detecting an embedded watermark in images, we consider the experimental results of the evaluation of the fractal dimension after

**Table 3** shows the results of measuring the fractal dimension of color fractal images in the form of a Julia set when embedding watermark by various known

The results show that the steganographic introduction of data does not actually affect the value of the fractal dimension, which makes it impossible to illegally

> **Embedding method**

300 1.5 LSB 1024 2.318 2.318

3000 LSB 2.109 2.109

**Image size**

Block based 2.318 2.317 Koch R 2.318 2.316 Koch G 2.318 2.316 Koch B 2.318 2.318

LSB 2.438 2.438 Block based 2.438 2.437 Koch R 2.438 2.436 Koch G 2.438 2.436 Koch B 2.438 2.438

Block based 2.109 2.115 Koch R 2.109 2.136 Koch G 2.109 2.097 Koch B 2.109 2.112

Block based 2.192 2.194 Koch R 2.192 2.208 Koch G 2.192 2.178 Koch B 2.192 2.195

Block based 2.172 2.179 Koch R 2.172 2.186

0.5 LSB 2.192 2.192

0.0005 LSB 2.172 2.172

**D of original fractal**

**D of fractal with watermark**

does not exceed 3%.

*Fractal Analysis - Selected Examples*

steganographic methods.

extract the watermark.

**Number of iterations**

**Starting point** *c*

0.7778+ 0.1316i

0.74543 +0.11301i

**98**

*Changing the value of the fractal dimension of images during steganographic embedding.*
