DIFFERENT CHAOTIC EQUATIONS USED IN DIGITAL WATERMARKING

Presented by:Debasis Sahoo

Roll No : 1155012M.TECH (CS & IS)

Guided by :Mr. Chittaranjan Pradhan, Asst. Professer

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Contents• Introduction

• Digital Watermarking

• Chaos sequence

• Arnold’s Cat Map

• Cross Chaotic

• Chaotic Baker Map

• Henon Chaos System

• Ikeda map

• Motivation

• Problem Statement

• Objective

• Work plan

• References2

IntroductionWatermarking is a branch of information embedding which is used to embed important information in digital media like images, photographs, digital music, or digital video.

Content providers want to embed watermarks in their multimedia objects (digital content) for several reasons like copyright protection, content authentication, tamper detection etc.

However, Digital image watermarking is the process of embedding an identifying piece of data in a digital image. In this study, a secure watermarking scheme for images is done, based on a chaotic equations.

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Introduction (Contd.) The chaos can be broadly defined as, the randomized or

scrambled the objects . Accroding to James Gleick commented,

“Over the last decade, physicists, biologists, astronomers, and economists have created a new way of understanding the growth of complexity in nature. This new science, called chaos, offers a way if seeing order and pattern where formerly only the random, erratic, the unpredictable – in short, the chaotic – had been observed [2].

Images As matter is composed of discrete units called atoms (which are

themselves composed of discrete units), so too images are composed of discrete units called pixels. A pixel is a small square representing some color value, which when taken together form the mosaic that is the image. The image is a m x n matrix, where m represents the number of rows of pixels and n the number of columns of pixels, with each entry in the matrix being a numeric value that represents a given color [2].

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Copyright Protection

Content Archiving

Meta-data Insertion

Broadcast Monitoring

Tamper Detection

Watermarking Applications

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Requirements Of Digital Watermarking

There are three main requirements of digital watermarking. They are transparency, robustness, and capacity.

A. Transparency or Fidelity

The digital watermark should not affect the quality of the original image after it is watermarked. According to Cox et al. (2002) transparency or fidelity is defined as “perceptual similarity between the original and the watermarked versions of the cover work”[1].

B. Robustness

Cox et al. (2002) defines robustness as the “ability to detect the watermark after common signal processing operations”. Watermarks should be robust against variety of such attacks[1].

 C. Capacity or Data Payload

Cox et al. (2002) define capacity or data payload as “the number of bits a watermark encodes within a unit of time or work”. This property describes how much data should be embedded as a watermark to successfully detect during extraction. Watermark should be able to carry enough information to represent the uniqueness of the image[1]. 6

Watermarks And Watermark Detection

A. Pseudo-Random Gaussian Sequence

A Gaussian sequence watermark is a sequence of numbers comprising 1 and -1 and which has equal number of 1’s and -1’s is termed as a watermark. It is termed as a watermark with zero mean and one variation. Such watermarks are used for objective detection using a correlation measure [1].

B. Binary Image or Grey Scale Image Watermarks

Watermarking algorithms embed meaningful data in form of a logo image instead of a pseudo-random Gaussian sequence. Such watermarks are termed as binary image watermarks or grey scale watermarks. Such watermarks are used for subjective detection [1].

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Binary conversion of a image

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Chaos sequence

• Chaos signals are a kind of pseudorandom, irreversible and dynamical signals, which poses good characteristics of pseudorandom sequences. Chaotic systems are highly sensitive to initial parameters. The output sequence has good randomness, correlation, complexity and is similar to white noise. Chaotic sequence has high linear complexity and non predictability. The model here is chaos 1-D Logistic and is

shown in equation (1)

……..(1)

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Arnold’s Cat Map

We have been started the chaotic equations from the famous Russian mathematician Vladimir I. Arnold as he had suggested that how the image can be scrambled using the chaotic equations. That’s why he had taken the image map of cat and done the experiment which is known as “Arnold’s Cat Map”.

An image (not necessarily a cat) is hit with a transformation that apparently randomizes the original organization of its pixels. However, if it is iterated enough times, as though by magic, the original image reappears [2].

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Fig .1 Encryption Using Chaos Sequences

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Chaotic maps• Arnold’s cat map ,

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Fig .2: Arnold’s Cat MAP [2]

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Fig.3 Arnold Transform On Image[2]

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After fifteen iterations, the pixel – as would any other pixel in the image – has returned to its initial position, and it would continue eternally along this circle if iterated accordingly. This agrees with the earlier observation that the 124 × 124 image has a period of fifteen[2].

Due to a less number of secret keys and repeatability of original image Arnold catmap be used for security system alone. So, for enhancing the security it requires further processing. But it is easy to implement , as well as after some iteration we get the original image

 

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Cross Chaotic

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Experimental Results

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Experimental Values 0.9980 0.2997 1.1182 1.1282 0.3004 -0.1552 0.8644 0.2203 1.1354 0.5092 0.1389

0.6839 0.3899 0.8482 0.3539 1.0980 1.0523 0.1779 0.1425 0.5240 0.8773 0.5205

0.3637 0.4426 1.0525 0.7108 -0.0728 1.0782 0.2250 1.2007 1.2331 0.4984 -0.2049

1.4605 1.5671 1.1802 0.8288 0.4296 -0.1031 -0.0058 0.5740 0.7805 0.4498 0.7443

0.6723 0.0598 0.9272 0.2564 1.3019 1.0521 0.2234 -0.0766 0.7631 0.3236 0.9599

0.4214 0.6576 0.6135 0.2267 0.6352 0.4545 0.6696 0.3374 1.1375 0.6690 -0.0768

1.0288 0.1945 1.3386 1.1465 0.3852 -0.2404 1.3296 0.6239 -0.1395 1.2189 0.4028

0.2232 0.5157 0.8441 0.4471 0.6886 0.6214 0.1896 0.7019 0.3485 1.0036 0.4929

0.3318 0.4557 1.0176 0.6622 -0.0288 0.9728 0.2120 1.3286 0.9968 0.1511 0.1060

0.5449 0.8170 0.4489 0.7099 0.6400 0.1354 0.8066 0.2731 1.2279 0.7343 -0.1049

and so on…..20

Binary conversion of the experimental values

1 0 1 1 0 0 1 0 1 1 0

1 0 1 0 1 1 0 0 0 1 1

0 0 1 1 0 1 0 1 1 0 0

1 1 1 1 0 0 0 1 1 0 1

1 0 1 0 1 1 0 0 1 0 1

and so on…

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Pixel Substitution using the Ikeda map

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Proposed Model

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Motivation• Here first we discuss about chaos sequence, in the 1-D logistic

equation the value of x(n ) always depends on mu, if it lies in between [3,4] we get the better results. After that we move to the Arnold’s transformation which is a part of a chaotic map, but in Arnold’s transformation less number of secret keys are used. That’s why the may get the information after some iteration. Though ,it have some disadvantages but it is easy to encrypt the information in an image as it is a one dimensional equation.

 

• To overcome the difficulties of Arnold transformation, we move to the cross chaotic where numbers of keys are more than the previous equations and these keys are generally the number unknown parameters which are used in cross chaotic. Similarly, in Ikeda map has numbers of unknown parameters which are fulfilled good robustness of image watermarking.

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Problem Statement

• Here techniques are used for encrypting the watermark image i.e. logistic chaos sequence and Arnold’s transformation. After extracting the watermark image, it may be easier for the intruder to decrypt the original watermark image as there are only two secret keys are used in chaos encryption and one secret key in Arnolds transformation. Hence, we need to give more security to the watermark image for encryption, so that it will be difficult for the intruder to extract the original image.  

• This method cannot resist attacks like Gaussian noise attack and median filtering attacks. That’s why we need to embed the watermark image more securely into the original image. So that it will be more difficult to extract the watermark image.

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Objective 

1. To give more security to the watermark image by encryption so that it will be difficult for the intruder to decrypt the watermark image. Here the sequence can be generated by cross chaotic map instead of chaos 1D logistic map to encrypt the image more securely. [8]

2. Similarly at the time of embedding, a more strong embedding technique can be used so that it will be difficult to extract the encrypted watermark image.

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Work Plan

• Stage 1

Discussion of chaos sequence method for encrypting the watermark image and embedding into original image.

• Stage 2 

Analysis of Arnold’s Transformation.

• Stage 3

Analysis of Cross chaotic map.

• Stage 4

Analysis of Ikeda map techniques.

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Conclusion

The watermarking is based on Arnold &Chaos squences. The pseudo-random sequence generated by Arnold and chaos system possesses feature of very high randomness, so the watermark image becomes more secure. But as we have discussed earlier that cross chaotic map [8] is sensitive to the secret keys, it has larger key space, and it may give better encrypted image if we will apply instead of chaos sequence.

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References[1] Vidyasagar M. Potdar, Song Han, Elizabeth Chang, “A Survey of Digital Image

Watermarking Techniques”, School of Information Systems, Curtin University of Technology, Perth, Western Australia ,2005 IEEE, p.p 1-2.

 

[2] Gabriel Peterson, “Arnold’s Cat Map”, Math 45 – Linear Algebra,Fall 1997, p.p 1-7.

 

[3] YU Honglei, WU Guang-shou, WANG Ting,LI Diantao1, YangJun1,MU Weitao, FENG Yu,YI Shaolei, MU Yuankao, “An Image Encryption Algorithm Based on Two Dimensional Baker Map”, Second International Conference on Intelligent Computation Technology and Automation, 2009, p.p 536-539.

 

[4] Zheng-Wei Shen, Wei-Wei Liao, Ya-Nan , “Blind Watermarking Algorithm Based On Henon Chaos System And Lifting Scheme Wavelet”, Department Of Mathematics, University Of Science And Technology Of Beijing 100083, China, 2009, p.p 308- 313.

 

[5] I Cox, M Miller , J Bloom , J Fridrich , T Kalker , “Digital Watermarking and

Steganography”, Second Edition. Elsevier, 2008 .30

• [6] Xiaogang Jia, “Image Encryption using the Ikeda map”, Engineering College of Armed Police Force of China XiAn, China, International Conference on Intelligent Computing and Cognitive Informatics. 2010.

• [7] Qiang Wang, Qun Ding, Zhong Zhang, Lina Ding, “Digital Image Encryption Research Based on DWT and Chaos”, Fourth International Conference on Natural Computation, 2008, pages.494-498.

• [8] Kuldeep Singh , Komalpreet Kaur ”Image Encryption using Chaotic Maps and

DNA Addition Operation and Noise Effects on it”, IEEE,2011, pages 17-24.

• [9] Yue Sun, Guangyi Wang “An Image Encryption Scheme Based on Modified Logistic Map”, School of Electronics Information Hangzhou Dianzi University Hangzhou, China, Fourth International Workshop on Chaos-Fractals Theories and Applications, 2011.

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Thank You!!!