Kutter-Jordan-Bossen Method

The Kutter-Jordan-Bossen method is a steganographic method that implements a digital watermark in an image. The method was introduced by Martin Kutter, Frederick Jordan and Frank Bossen in the Journal of Electronic Imaging in April 1998 ^[1] .

The Kutter-Jordan-Bossen method belongs to the class of algorithms that hide data in the spatial domain. In the algorithms of this class, the implementation of the CEH is carried out by changing the brightness or color components of the pixel . In this method, individual watermark bits are repeatedly embedded in the image by changing the value of the blue channel in the pixel. This change is proportional to the brightness component of the pixel and can take both positive and negative values depending on the value of the embedded watermark bit ^[2] .

The main properties that the CEH should have are indistinguishability for the human eye and resistance to various distortions and image changes. The Kutter-Jordan-Bossen method satisfies the first requirement by embedding the watermark bits in the blue pixel channel, since the human eye is the least sensitive to this color ^[3] . Resistance to image distortion is provided by repeatedly embedding the CEH bits in various parts of the original image.

Scientific works in the field of application of this method have shown high efficiency when it is used to protect the copyright of images and video materials ^[4] ^[5] (see Application of steganography ), as well as to verify the integrity of images and QR codes ^[6] .

Content

Algorithm

Embedding one bit of information

Let be $s$ ${\ displaystyle s}$ - a separate bit embedded in the container $I=\{R,G,B\}$ ${\ displaystyle I = \ {R, G, B \}}$ information where $R$ ${\ displaystyle R}$ , $G$ ${\ displaystyle G}$ and $B$ ${\ displaystyle B}$ - red, green and blue components of the pixel, respectively, and $p=(i,j)$ ${\ displaystyle p = (i, j)}$ - pseudo-random position with coordinates $i$ ${\ displaystyle i}$ and $j$ ${\ displaystyle j}$ in which the attachment is made. The pixel position is determined by the key. $K$ ${\ displaystyle K}$ , which is used as the initial state ( English seed) in the pseudo random number generator . Bit $s$ ${\ displaystyle s}$ embedded in the image by changing the value of the blue channel in the pixel with the position $p$ ${\ displaystyle p}$ in proportion to the luminance component of the pixel $L=0.299R+0.587G+0.114B$ ${\ displaystyle L = 0.299R + 0.587G + 0.114B}$ . So instead of the blue component $B_{ij}$ ${\ displaystyle B_ {ij}}$ a new value is calculated, calculated by the formula $B_{ij}^{*}=B_{ij}+(2s-1)L_{ij}q$ ${\ displaystyle B_ {ij} ^ {*} = B_ {ij} + (2s-1) L_ {ij} q}$ where $q$ ${\ displaystyle q}$ Is a constant characterizing the energy of the embedded signal. Its value depends on the purpose of the circuit. More than $q$ ${\ displaystyle q}$ , the higher the robustness (stability) of the investment, but the greater its visibility ^[7] .

Extracting one bit of information

The recipient extracts the watermark information bits without the presence of the original image. In order to restore the original bit, it is necessary to predict its value. As such a prediction, a linear combination of blue channel values of adjacent bits is used. The authors of ^[8] empirically established that the use of neighbors from a cruciform neighborhood of a pixel (pixels with the same coordinate $i$ ${\ displaystyle i}$ or $j$ ${\ displaystyle j}$ relative to the original pixel) give the best prediction. Therefore, the predicted value ${\hat {B}}_{ij}$ ${\ displaystyle {\ hat {B}} _ {ij}}$ can be calculated by the formula:

{\hat {B}}_{ij}={\frac {1}{4c}}\left(\sum _{k=-c}^{c}B_{i+k,j}+\sum _{k=-c}^{c}B_{i,j+k}-2B_{ij}\right)

{\ displaystyle {\ hat {B}} _ {ij} = {\ frac {1} {4c}} \ left (\ sum _ {k = -c} ^ {c} B_ {i + k, j} + \ sum _ {k = -c} ^ {c} B_ {i, j + k} -2B_ {ij} \ right)}

,

Where $c$ ${\ displaystyle c}$ - dimensions of the cross-shaped neighborhood of the pixel on top (bottom, left, right). To recover the value of the CEH bit, the difference is calculated $\delta =B_{ij}-{\hat {B}}_{ij}$ ${\ displaystyle \ delta = B_ {ij} - {\ hat {B}} _ {ij}}$ . Sign of difference $\delta$ ${\ displaystyle \ delta}$ just determines the value of the built-in bit of information ^[8] .

Although the correct extraction is most likely, it is not guaranteed. The functions of embedding and extracting information from the container are not symmetrical, that is, the extraction function is not inverse to the embedding function, therefore, the value of the predicted bit is probabilistic and, generally speaking, may not coincide with the original value of the bit of the digital watermark. In order to increase the likelihood of correctly extracting a bit of information, multiple embeddings of the same bit into a container is used ^[8] .

Embedding a bit of information repeatedly

To increase the likelihood of correct restoration of the CEH in the Kutter-Jordan-Bossen method, multiple embedding of an information bit in $n$ ${\ displaystyle n}$ container positions. These $n$ ${\ displaystyle n}$ positions $p_{1},...,p_{n}$ ${\ displaystyle p_ {1}, ..., p_ {n}}$ are determined by a pseudo-random sequence . As before, the initial state of the pseudo-random number generator is determined by the key $K$ ${\ displaystyle K}$ . Redundancy of information is determined by the density parameter $\rho$ ${\ displaystyle \ rho}$ . Density determines the likelihood that a particular pixel will be used to embed information. The value of this parameter lies in the range from 0 to 1, where 0 means the complete absence of embedding information, and 1 means embedding the CEH bits in each pixel ^[8] .

Image Crawl Order in the Kutter-Jordan-Bossen Method

The pixel coordinates for embedding are defined as follows: for each pixel in the image, a pseudo-random number is generated $x$ ${\ displaystyle x}$ . If a $x<\rho$ ${\ displaystyle x <\ rho}$ then the bit of information should be embedded in the pixel. Otherwise, the pixel remains untouched. In this case, the pixels of the image are bypassed in a zigzag fashion, instead of iterating over the coordinates of the pixel row by row (or column by column). This is done so that the procedure for embedding information is independent of image size ^[8] .

In this method, a pseudo-random number generator is used only to determine pixel positions. Since not all pixels are subject to change and there is no explicit way to determine which pixels were changed, the pseudo random number generator does not have to be cryptographically robust ^[8] .

As before, to recover a bit, the difference between the predicted and the actual bit values for each $p_{k}$ ${\ displaystyle p_ {k}}$ : $\delta _{k}=B_{p_{k}}-{\hat {B}}_{p_{k}}$ ${\ displaystyle \ delta _ {k} = B_ {p_ {k}} - {\ hat {B}} _ {p_ {k}}}$ ^[8] .

Then the differences obtained are averaged : ${\bar {\delta }}={\frac {1}{\rho |I|}}\sum _{k}\delta _{k}$ ${\ displaystyle {\ bar {\ delta}} = {\ frac {1} {\ rho | I |}} \ sum _ {k} \ delta _ {k}}$ where $|I|$ ${\ displaystyle | I |}$ - the number of pixels in the image $I$ ${\ displaystyle I}$ . Sign of the averaged difference $\delta _{k}$ ${\ displaystyle \ delta _ {k}}$ determines the value of the embedded information bit ^[8] .

Embedding an m- bit digital watermark

Embedding $m$ ${\ displaystyle m}$ bit digital watermark $S=\{s_{0},...,s_{m-1}\}$ ${\ displaystyle S = \ {s_ {0}, ..., s_ {m-1} \}}$ carried out as follows. Let be $p_{1}...p_{n}$ ${\ displaystyle p_ {1} ... p_ {n}}$ - positions selected for multiple embedding of one CEH bit. Then, for each of these positions, a bit of information is selected and embedded. When embedding a message of length $m-2$ ${\ displaystyle m-2}$ bit, two extra bits are added. These two bits are always selected equal to 0 and 1, respectively. This improves the probability of extracting information by determining a threshold value. $\tau$ ${\ displaystyle \ tau}$ (see next subsection). Also, these bits determine the geometric location, which is used to resist geometric attacks, such as rotation and cropping of the image ^[9] .

Steganographic Resistance

Adaptive Threshold

In ^{[9], the} authors of this method provide an analysis of the stability of the algorithm to steganographic attacks . The authors constructed graphs of the dependence of the value of the number ${\bar {\delta }}^{b}$ ${\ displaystyle {\ bar {\ delta}} ^ {b}}$ (difference between true and predicted bit values) from the bit number $b$ ${\ displaystyle b}$ for two images: images with the CEH built-in using the Kutter-Jordan-Bossen method, and images after attempting a steganographic attack. In the original image with the CEH mark ${\bar {\delta }}^{b}$ ${\ displaystyle {\ bar {\ delta}} ^ {b}}$ can be uniquely determined, while after a steganographic attack, the determination of the sign ${\bar {\delta }}^{b}$ ${\ displaystyle {\ bar {\ delta}} ^ {b}}$ is not clear enough. The solution to this problem is to introduce the previously mentioned threshold value. Since it is known that the first two bits of information have values 0 and 1, respectively, it is possible to use this information to calculate the adaptive threshold value $\tau$ ${\ displaystyle \ tau}$ . It is calculated as the average ${\bar {\delta }}^{0}$ ${\ displaystyle {\ bar {\ delta}} ^ {0}}$ and ${\bar {\delta }}^{1}$ ${\ displaystyle {\ bar {\ delta}} ^ {1}}$ :

{\bar {s}}^{b}={\begin{cases}1,&{\bar {\delta }}^{b}>{\frac {{\bar {\delta }}^{0}+{\bar {\delta }}^{1}}{2}}\\0,&{\bar {\delta }}^{b}\leq {\frac {{\bar {\delta }}^{0}+{\bar {\delta }}^{1}}{2}}\end{cases}}

{\ displaystyle {\ bar {s}} ^ {b} = {\ begin {cases} 1, & {\ bar {\ delta}} ^ {b}> {\ frac {{\ bar {\ delta}} ^ {0} + {\ bar {\ delta}} ^ {1}} {2}} \\ 0, & {\ bar {\ delta}} ^ {b} \ leq {\ frac {{\ bar {\ delta }} ^ {0} + {\ bar {\ delta}} ^ {1}} {2}} \ end {cases}}}

The idea of using an adaptive threshold value is based on the assumption that changes made during a steganographic attack equally affect all embedded bits in the image. This assumption can be made since each bit of the digital watermark is embedded several times and evenly distributed throughout the image. Therefore, any changes made over the image will equally affect all bits of the restored CEH, assuming that the number of embeddings is quite large ^[9] .

Notes

↑ Digital watermarking of color images using amplitude modulation, 1998 .
↑ Digital watermarking of color images using amplitude modulation, 1998 , p. 326.
↑ Steganographic systems. Criteria and methodological support: Teaching aid, 2016 , p. 25-29.
↑ Multimedia Watermarking Techniques, 1999 , p. 1087.
↑ Applying of Kutter-Jordan-Bossen steganographic algorithm in video sequences, 2017 , p. 696.
↑ Digital watermarking for QR-code protection, 2014 , p. 121-122.
↑ Digital Steganography, 2002 , p. 164.
↑ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Digital watermarking of color images using amplitude modulation, 1998 , p. 327.
↑ ¹ ² ³ Digital watermarking of color images using amplitude modulation, 1998 , p. 328.

Literature

Gribinin V.G., Kostyukov V.E., Martynov A.P., Nikolaev D.B., Fomchenko V.N. Steganographic systems. Criteria and methodological support: Teaching aid / Edited by Doctor of Technical Sciences V. G. Gribunin. - Sarov: FSUE RFNC-VNIIEF, 2016. - 324 p. - ISBN 978-5-9515-0317-6 .

Kutter M., Jordan F., Bossen F. Digital watermarking of color images using amplitude modulation (English) : Journal. - 1998 .-- 1 April ( vol. 7 , no. 2 ). - DOI : 10.1117 / 1.482648 .

Hartung F., Kutter M. Multimedia Watermarking Techniques (English) // Proceedings of the IEEE: Journal. - 1999 .-- July ( vol. 87 , no. 7 ).

Lysenko N., Labkov G. Applying of Kutter-Jordan-Bossen steganographic algorithm in video sequences (Eng.) // IEEE: Journal. - 2017 .-- April 17. - DOI : 10.1109 / EIConRus.2017.7910651 .

Narimanova O.V., Semenchenko D.M. Digital watermarking for QR-code protection (in Ukrainian) // Politekperiodika (Odessa): journal. - 2014. - T. 1 , No. 15 . - ISSN 2308-8060 .

Gribinin V.G., Okov I.N., Turintsev I.V. Digital steganography. - Solon-Press, 2002 .-- 235 p. - ISBN 5-98003-011-5 .

[_6425f863c207dc31-1] Digital watermarking of color images using amplitude modulation, 1998 .

[_ee99a4aedc71bfb0-2] Digital watermarking of color images using amplitude modulation, 1998 , p. 326.

[_ddd97182b1c6edf0-3] Steganographic systems. Criteria and methodological support: Teaching aid, 2016 , p. 25-29.

[_12ca302e53aebbbf-4] Multimedia Watermarking Techniques, 1999 , p. 1087.

[_ed5decb407af6a9c-5] Applying of Kutter-Jordan-Bossen steganographic algorithm in video sequences, 2017 , p. 696.

[_0f1bdfb2628b2829-6] Digital watermarking for QR-code protection, 2014 , p. 121-122.

[_cbb7f883742b094d-7] Digital Steganography, 2002 , p. 164.

[_ee99a4aedc71bfb1-8] ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ Digital watermarking of color images using amplitude modulation, 1998 , p. 327.

[_ee99a4aedc71bfbe-9] ¹ ² ³ Digital watermarking of color images using amplitude modulation, 1998 , p. 328.