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Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego



Rengarajan Amirtharajan and John Bosco Balagurn Rayappan
 
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ABSTRACT

Computers have invaded all premises of the human world, starting from a grocery store to a missile launching center. Because of the omnipresence of computers, it becomes more and more difficult everyday to secure the confidential information from misuse. The fairly common technique of cryptography has been proved inadequate in recent years. Steganography, a contemporary yet an age-old technique to hide secret data into an unsuspected cover media like an image, thereby preventing the recognition of the very presence of secret data, is an alternative. In this study, an improved image steganographic approach is proposed. This method reduces the mean square error (MSE) by localizing the error-reduction process to every row. The error reduction is performed by selective embedding of the actual secret, its binary complement, gray-coded version or inverted gray-coded version. Of the four versions, the version giving the least MSE is embedded on a row-by-row basis. This method reduces the MSE by a factor of 1.8 and boosts the peak signal to noise ratio (PSNR) by a 0.25 db and considerably increases the security.

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  How to cite this article:

Rengarajan Amirtharajan and John Bosco Balagurn Rayappan, 2012. Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego. Journal of Applied Sciences, 12: 428-439.

DOI: 10.3923/jas.2012.428.439

URL: https://scialert.net/abstract/?doi=jas.2012.428.439
 
Received: December 12, 2011; Accepted: January 27, 2012; Published: April 03, 2012



INTRODUCTION

Power has taken several incarnations ever since its genesis. Right from Stone Age to the present day, power has kept its possessor at the summit of hegemony and thus it is the most sought after commodity. In the electronic epoch power has manifested itself in the form of classified and critical information. Since the human race has succumbed to enticing power to such an extent that iniquity today is skyrocketing, there is a need to protect information from falling into the wrong hands and to prevent clandestine and unscrupulous activities. Providentially, the advancements in technology have begot many techniques to maintain the veracity and variability of the crucial information giving rise to an entire discipline called information hiding. Information hiding is stratified into several subsets namely cryptography (Schneier, 2007), steganography and watermarking (Stefan and Fabin, 2000; Zaidan et al., 2010).

Cryptography (Schneier, 2007) is the art of writing esoteric information in an occult fashion thereby rendering it scrutable only to the authorized receiver. In contrast to cryptography which focuses on keeping the contents of a message secret, steganography (Stefan and Fabin, 2000; Zaidan et al., 2010) focuses on keeping the very existence of a message secret. Steganography is implemented in digital audio (Zhu et al., 2011), video (Al-Frajat et al., 2010) and images (Amirtharajan and Balaguru, 2009, 2010, 2011; Amirtharajan et al., 2012; Bender et al., 1996) of which image steganography has gained much appreciation and commendation in the recent past. In image steganography the vital information is dissembled in a cover image with assiduous efforts resulting in a stego image. The embedded secret information is imperceptible to the human eye thereby rendering the image impregnable (Yang, 2008).

In the available literature many researchers proposed an assortment of approaches to information hiding. These methodologies have different characteristics like capacity, imperceptibility and robustness (Amirtharajan and Balaguru, 2009, 2010, 2011; Kumar et al., 2011). These characteristic are inevitable for different applications, such as secret communication (Stefan and Fabin, 2000), copyright protection (Wang and Lin, 2004; Yen and Tsai, 2008) and tampering detection or integrity check (Lin et al., 2005).

Information hiding techniques could be categorized into two types: methods in the spatial domain and methods in the frequency domain. In the spatial domain approach, the secret messages are embedded by directly injecting secret data in the image pixels (Chan and Cheng, 2001, 2004; Wang et al., 2001; Chang et al., 2003; Yang, 2008; Thien and Lin, 2003). Whereas in the later case, the frequency domain approach the image is first transformed into its frequency domain (Amirtharajan and Rayappan, 2012a,b; Chang et al., 2002) then the secret messages are embedded in the transformed coefficients.

The major concern is about the objective of transmitting secret data, the stego method should possess high capacity, high quality and imperceptibility. More number of research papers have been intended for this theme and performs the embedding operations in the spatial domain either using raster scan or random scan (Amirtharajan and Balaguru, 2009, 2010; Amirtharajan et al., 2011, 2012; Yen and Lin, 2010). A detailed survey on Information hiding till 1999 is available by Petitcolas et al. (1999). A complete survey on image steganography could be found by Cheddad et al. (2010) and on random image steganography and steganalysis in Amirtharajan et al. (2012) three more survey on Field Programmable Gate Array (FPGA) for steganography, middle ware for cryptography/ steganography and Orthogonal Frequency Division Multiplexing (OFDM)+Code Division Multiple Access (CDMA)+stego for secure communication is available by Rajagopalan et al. (2012), Janakiraman et al. (2012a) and Thenmozhi et al. (2012), respectively. There are three kinds of approaches called LSB-based (Chan and Cheng, 2001, 2004; Wang et al., 2001; Chang et al., 2003; Yang, 2008; Thien and Lin, 2003; Amirtharajan and Balaguru, 2009, 2010), PVD-based (Wang et al., 2008; Amirtharajan et al., 2010) and mod-based (Chan and Cheng, 2004; Thien and Lin, 2003; Wang et al., 2008) are commonly available in literature and sometimes it could be combined to offer both capacity, imperceptibility and to improve the security (Chang et al., 2003; Hmood et al., 2010a, b; Xiang et al., 2011; Lin et al., 2005; Janakiraman et al., 2012b; Zaidan et al., 2010, 2011 and Zanganeh and Ibrahim, 2011). The counter attack on steganography called steganalysis are detailed (Xia et al., 2009; Qin et al., 2009). A detailed review on steganalysis is reported by Qin et al. (2010).

In LSB-based approaches, secret data are embedded by directly substituting the least-significant-bits (LSBs) with equal bits of the secret for each pixel. Furthermore, techniques based on pixel-value differencing (PVD) modify the difference value between a pair of pixels to fit the value of the embedded secret. Finally, mod based approaches which use the modular operation, are similar to k-bit LSB-based approaches if the modulus is 2k.

Motivated by this study, a simple and effective stego method has been proposed to improve the stego image quality and to introduce cryptic effect while embedding.

PRELIMINARY RELATED WORKS

Chan and Cheng (2004) proposed an LSB-based hiding scheme using an optimal pixel adjustment process (OPAP). Their method adjusts each pixel after the message is embedded to improve the quality of the stego object and their experimental results showed that their method yielded quicker results. Yang (2008) proposed new LSB-based approach, named as the Inverted pattern (IP) LSB substitution approach. Later this method combined with OPAP called IPLSB to improve the quality of the stego image. In this study, we have adapted a new LSB-based approach based on Yang (2008), named as the inverted pattern binary and gray (IPBG) LSB substitution approach, to further highlight the quality of the stego-image Before secret messages are embedded, some secret messages are transformed by inverting operation and some secret messages are not. A simple strategy is used to judge whether a section of messages is inverted and a bit string named as the IPKey is used to record these inverting actions. Also, we combine the concept of the OPAP with our approach to improve image quality further. The experimental results show that the proposed approach results in a better image quality than that of the optimal LSB substitution approach (Wang et al., 2001; Chan and Cheng, 2001; Thien and Lin, 2003), the OPAP LSB substitution approach (Chan and Cheng, 2004) and inverted pattern approach (Yang, 2008).

In a normal LSB substitution the RGB (red blue green) image is converted in to gray image and then last few least significant bits of gray image are selected according to key length k and the message which is to be embedded is converted to series of ASCII values of the characters in the message and then to binary. Message is then stored in the cover according to the method of embedding. The series of operations done in LSB substitution are as follows:

Let C be the original 8-bit grayscale cover-image of McxNc pixels represented as:

C = {xij| 0 ≤ I ≤ Mc, 0 ≤ j ≤ NC
xij ∈ :0, 1, 2,,,,, 225}

D be the n-bit secret data represented as:

D = {di |0 ≤ I ≤ n, di ∈ {0, 1}}

Suppose that the n-bit secret data Dd (decimal representation) is to be embedded into the k-rightmost LSBs of the cover-image C:

S = C-C mod 2k + Dd

Here, S is the Stego object, C cover object and Dd is the decimal equivalent of the secret data.

In the extraction process, given the stego-image S, the embedded messages can be readily extracted without referring to the original cover-image. The k LSBs of the selected pixels are extracted and lined up to reconstruct the secret message bits. Mathematically, the embedded message bits D can be recovered by:

Dd = S mod 2k

The OPAP simply improves the stego object after embedding the secret data, either by adding or subtracting 2k without affecting the rightmost k secret data bits in the stego cover.

THE PROPOSED METHOD

Embedding: A schematic diagram of the proposed method is given in Fig. 1 and 2. Initially the secret data or message is encrypted using Data Encryption Standard DES (Schneier, 2007), is a symmetric key cryptography algorithm. The cover image is split into separate rows. The order of rows considered for embedding data is chosen using a Pseudo random number generator with a chosen seed. For each row, a try is made to embed data, inverse data, gray code of data and the inverted gray code of data. The encoded form of the confidential information on the selected row which offers minimum Mean Square Error (MSE) is chosen and fixed for the same. This binary/ inverted binary/gray and inverted gray data pattern is stored as IPKEY. Thus, on an average MSE is reduced to a greater extent. The Stego image, IPKEY and the seed are communicated.

Mathematical model for row wise, inverted pattern LSB embedding

General formulae: 1’s complement of a No.:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(1)

Where:
k = No. of bits
x = Number to be inverted in bits
Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego = 1’s complement of the number

For example, take a 4 bit binary representation of a number ‘2’ [0010] as x here:

k = 4 so, Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego-2 = (16-1)-2
Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego = 13[1101] complement of [0010] ‘2’

LSB embedding:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(2)

Where:
k = No. of bits to be embedded
Ci = Cover pixel
Si = Stego pixel
mi = k – bit message block in decimal

For example, let k = 4. Ci = 16[0001000] and mi is ‘2’ [0010] as mi:

Si = 16- 16 mod 24 + 2 = 16-16mod16+2 = 18
Si = 8[00010010]

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Fig. 1: Proposed schematic diagram for embedding

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Fig. 2: Block diagram for extraction

LSB recovery:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(3)

where, symbols are same as Eq. 2:

Let Si = 18[00010010]
To Extract the last 4[since k = 4] bits, we have:

mi = 18 mod
24 = 18 mod
16 = 2 [0010]

General IDEAS
  Four flavours of secret data:

Plain data-m (i,j)
Inverted data - Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i, j)
Grey Coded data-g (i,j)
Inverted Grey Coded data-g' (i,j)

R rows: In each cover image, there are ‘R’ No. of rows, each of same length D, where:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(4)

where, the McxNc are dimensions of the cover image. Each row is denoted as ri, where, i∈N and I≤R i.e., Set of rows = {ri, ∀ i∈N and i≤R}

Each row ri is in turn a matrix, denoted as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(5)

In other words, each row has D pixels.

Message data (secret) to be embedded [k bit length]: m (i, j), where:

i = Row identifier
j = Pixel inside a row

The complement of m (i, j) is denoted as Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i, j)

Embedding procedure: Let the cover image be C with McxNc pixels.

Let it be divided into R blocks named ri, ri,……., riR, each having equal number of pixels D:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(6)

Also, ri = [ri1, ri2,……., riD], where i ∈ N and i ≤ R

Let ‘k’ be the number of LSBs to be replaced in cover pixels.

Let the secret message be a matrix M, where each elements of M is made up of k bits. Then we can denote the message to be embedded in the ith row, jth pixel as m (i, j). Let s (i, j) denote the stego value of jth pixel in the ith row, when message m (i, j) is embedded in cover pixel rij. Alternatively Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i, j) is embedded instead of m (i, j) then the stego pixel is denoted as Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i, j):

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(7)

If we consider R blocks of stego image as s1,s2,…………..sR. Then, si = s (i) or Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i) or sg (i) or sg' (i) where MSE is minimum and s (i) = {s (i, j), j∈N and j≤D}

key matrix is denoted as:

K = [K1,K2,……..KR]

where, Ki is chosen based on the following conditions 00-if m (i,j) is embedded:

01 = if Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i, j) is embedded
10 = if g (i,j) is embedded
11 = if g' (i,j) is embedded

Retrieval procedure: Key matrix is denoted as:

K = [K1,K2,……..KR]

The preliminary, unprocessed message mu (i,j) can be extracted from pixels in stego image as: mu (i,j) = s (i,j) mod 2k from Eq. (3) the actual message m (i,j) can be extracted by processing mu (i,j) as follows:

m (i,j) is chosen from the following conditions based on Ki value
mu (i,j) - if corresponding Ki = 00
(2k-1)-mu (i,j), if Ki = 01
g-1 (i,j), if Ki = 10 (if g-1 denotes inverse of grey code function)
g'-1 (i,j) = (2k-1)-g-1 (i,j) (if Ki = 11 )
(if g'-1 denotes inverse of inverted grey code function)

WORST CASE MSE

The worst case MSE for a block with D pixels is defined as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(8)

MSE for ith row, when m (i) (actual data) is embedded, is given as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

When inverted data Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (i) is embedded, then, MSE for the same parameters is denoted as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

When grey coded data g (i) is embedded, then MSE for the same parameters is denoted as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

When inverse grey coded data g’ (i) is embedded, then MSE for the same parameters is denoted as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

According to the embedding procedure, minimum MSE is chosen. The minimum MSE for a row is defined as:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

= D-1Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego (2k – 1)2 ( since sum of all s,Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego, sg and Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego components:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(9)

We know that, if any n numbers x1, x2, x3, x4,…, xn add up to produce a total T, then:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(10)

Thus, applying (10) in (9), we get:
MSEmin (i)≤(1/2) MSEw (i) for all I≤R

Thus, we get MSE for any block to be less than or equal to ½ of the worst case MSE.

Random k-bit Adaptive Embedding

Inputs:

Sampled Cover Image C
Secret data bit stream M
Key E for Encryption

Outputs:

Stego Image (S), containing embedded secret data
KEY (Used for recovery)

Algorithm for embedding:

Step 1: Encrypt the secret data (M) using DES (Data Encryption Standard) with key E
Step 2: Let P = length of secret data stream M (in number of bits) got from Step-1
Step 3: Split the cover image C into separate rows . Let N = Total number of rows
Step 4: Generate a array PRN of N pseudo-random numbers in the range [0,N-1] where each No. occurs only once. Let the seed be stored in a text file
Step 5: Invert the bit array M to give Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego. Encode the bit array M using Grey Code to give G and invert G to give Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Step 6: Let i = 1 (Here, i is the row counter)
Step 7: Select PRNG[i]th block and perform the following operations

Selective embedding
{a. Let r = pixel index array (for traversal)
b. For ( j = 1 to length (r) ) do (Here j is the pixel counter)
  {
Replace k LSBs of jth pixel of the selected block with k bits from M to give O[i,1]
  }
c. Compute MSE
d. For ( j = 1 to length (r) ) do (Here j is the pixel counter)
  {
Replace k LSBs of jth pixel of the selected block with k bits from Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego to give O[i,2]
  }
e. Compute Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
f. For ( j = 1 to length (r) ) do (Here j is the pixel counter)
  {
Replace k LSBs of jth pixel of the selected block with k bits from G to give O[i,3]
  }
g. Compute MSEGray
h. For ( j = 1 to length (r) ) do (Here j is the pixel counter)
  {
Replace k LSBs of jth pixel of the selected block with k bits from Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego to give O[i,4]
  }
g. Compute MSE gray
h. If MSE is greatest

KEY[i] = ”00”
Else if Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego is greatest
KEY[i] = ”01”
Else if MSEGray is greatest
KEY[i] = ”10”
Else
KEY[i] = ”11”
Assign MSE[i] = Minimum MSE
}

Choose STEG[i] as the value of O for which MSE is minimum
P = P-k. (Reduce length as k bits have been embedded)
If P>0 then assign i = i+1. Else, goto step-8 (that is check whether message is finished)
If i>N then goto step-8 (check whether EOF is reached for cover image)
Step 7: F. Goto
Step 8: Save the array STEG as the stego image array S
Step 9: Save S into a image file and KEY in a text file
Step 10: Communicate S,KEY and seed used to generate PRN

Recovery process: The same Pseudo random number sequence is generated using the received seed. Using the KEY, the pattern is identified for different rows. Recovery modules are run to recover the secret. The result is then decrypted using DES to get the message back.

Random k-bit adaptive recovery

Inputs:

Stego Image (S), containing embedded secret data
Key E for decryption. KEY in text file from embedding process
Seed (to generate Pseudo Random Number Generator PRNG)

Output:

Secret data bit stream M

Algorithm for extraction

Step 1: Split the stego image S into separate rows. Let N = Total number of rows
Step 2: Generate a array PRNG of N pseudo-random numbers in the range [0,N-1] where each No. occurs only once
Step 3: Let i = 1 (Here, i is the row counter)
Step 4: Select PRNG[i]th row and perform the following operations:

Get Message M using retrieval
B. If KEY[i,1] = ”01”
M[i] = Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego [I]
Else if KEY[i,1] = ”10”
M[i] = MGray[i]
Else if KEY[i,1] = ”11”
M[i] = Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Else
M[i] = M[i]

Assign i = i+1 (increment row count)
If i>N goto step-5 else goto Step-4
   
Step 5: Decrypt M using DES and write it to text file as output

RESULTS AND DISCUSSION

In this present implementation Lena, Baboon, Gandhi and Temple 256x256 pixel Images have been considered by varying k = 1, 2, 3 and 4 bit LSB embedding, then stego image quality has been improved with OPAP. The effectiveness of the proposed system has been estimated by computing the MSE and PSNR of the Stego object with cover object.

The MSE is calculated by using the equation:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(11)

where, M and N denote the total number of pixels in the horizontal and the vertical dimensions of the image Xi,j represents the pixels in the original image and Yi, j, represents the pixels of the stego-image.

The Peak Signal to Noise Ratio (PSNR) is calculated using the equation:

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
(12)

where, Imax is the intensity value of each pixel which is equal to 255 for 8 bit gray scale images. Higher the values of PSNR better the image quality. The cover image is given in Fig. 3a and the corresponding stego images for k = 1 in Fig. 3b, k = 2 in Fig. 3c, k = 3 in Fig. 3d, k = 4 in Fig. 3f and the proposed stego results in Fig. 3g for k = 4, MSE, PSNR is given in Fig. 4 and 5, respectively.

In the case of simple LSB embedding for full embedding capacities 256x256 bits for k = 1, 256x256x2 bits for k = 2 and so on. While using secret data in binary format alone for k = 4 the MSE is 36.60, inverted binary is 36.90, gray is 41.17, inverted gray is 40.88 and the proposed is 34.84. The proposed method MSE by adapting quantum of 64 pixels further reduces it to 32.81. These values are shown in Table 1- 3.

The corresponding PSNR value of the proposed method improved to 35.03826 dB which is for better than Chan and Cheng (2004) method PSNR of 34.8 dB. The corresponding MSE value of the proposed method reduced to 20.38 which is for better than Chan and Cheng (2004) method MSE of 21.6 and Yang (2008).

Image steganography is successfully implemented using a novel encoding method in which various bit representations namely binary, inverted binary, gray and inverted gray are employed.

Table 1: Comparative MSE values for Full Embedding capacity on Lena and Baboon by splitting into 256 pixels as one block
Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Fig. 3(a-g): (a) Cover Images Lena, Baboon, Gandhi and Temple, (b) k = 1, (c) k = 2, (d) k = 3, (e) k = 4, (f) k = 4 Proposed 256 Stego Images Lena, Baboon, Gandhi and Temple and (g) k = 4 Proposed 64 Stego Images Lena, Baboon, Gandhi and Temple

Table 2: Comparative MSE values for full embedding capacity on Lena and Baboon by splitting into 64 pixels as one block
Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

Table 3: Comparison of MSE values with other methods for full embedding capacity in Lena, Baboon, Gandhi and Temple
Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Fig. 4: Comparative MSE values for full embedding capacity on Lena

Here the secret data, encoded in all the four representations is embedded in a row of the cover image and the MSE is calculated exclusively for each of the four encoding bit representations. Of the representations the one that yields the least MSE is adopted for the respective row. In this way all the four forms of representation are used in each row and the form resulting in the least MSE and PSNR is espoused and the results are given in Fig. 4 and 5.

Finally a key is formulated using a code to depict the bit representation format employed in each row which again is arcane thereby protecting the stego image from malicious aggressors.

Image for - Brownian Motion of Binary and Gray-Binary and Gray Bits in Image for Stego
Fig. 5: Comparative PSNR values for full embedding capacity on Lena

Considering an image of dimensions 256x256, key bits per row is 2. Therefore, in order to account for 256 rows, we get 256x2 = 512 bits. These 512 bits of data form a secret key array. Thus, we can define key-to-data ratio as 512/(256x256x8) = 512/ (65536x8) = 0.00098 = 0.098%. Furthermore one more experiment has been carried out to improve the quality of the stego image by splitting each row into 4 quantum units of 64 pixels. The results are encouraging with slight increase in the key length.

Since the embedding depends upon the Least Mean Square Error which is dynamically determined by the combination of cover image pixels and secret data bits, any attack to recover the data without using the secure key becomes impossible.

COMPLEXITY ANALYSIS

The DES cryptography system introduces a complexity of 264
For 256*256 pixel image, total number of rows will be 256
These 256 rows can be selected in a random manner in 256! Ways
In each row one embedding technique out of four is chosen
If we are embedding k bits in each pixel then
The total complexity = 264 *256!* 4*8/k
So total complexity in this case will be 264 *256!* 4 *8/k

For proposed 64 method:

The DES cryptography system introduces a complexity of 264
For 256*256 pixel image, total number of rows will be 256
These 256 rows can be selected in a random manner in 256! Ways
Each row is grouped into 4 blocks of 64 pixels.
For each block a particular technique is selected out of 4
If we are embedding k bits in each pixel then
Total complexity is 264 *256!* 4 *8/k
If we select the blocks in each row in a random manner, there will be 4 blocks and we can select it in 4! Ways and if we select the pixels in a block in a random manner then
Total complexity is 264 *256!* 4 *4!*8/k
This security level estimation reveals the of the proposed stego against hackers

CONCLUSION

By simultaneously serving two ultimate requirements of security, i.e., greater imperceptibility (least MSE) and high complexity (cryptic effect created by the choice of row-wise embedding), the proposed technique promises un-tampered transmission and authorized use of secret data. Usage of nominal key length reduces the cost associated with the transport of key over a secure channel. To summarize the key points in this paper:

An improved image steganographic method has been proposed, implemented and tested called Brownian motion of Binary and Gray-Binary and Gray Bits in Image for stego
It is a variation of Yang (2008) method with additional choices of Gray code and inverted Gray code along with binary and inverted binary. This provides four choices for the data to be embedded. Thus, it further reduces the effective Mean Square Error to make the stego image more imperceptible and also gives cryptic effect
A mathematical model has been developed to justify the work
The worst case Mean square error is derived MSEProposed≤ (1/2) MSEwLSB and the results are discussed in detail
This method reduces the MSE by a factor of 1.8, without compromising the data embedding capacity and marginal improvement in imperceptibility. (In Information hiding with respect to magic triangle capacity, imperceptibility and Robustness). The proposed method will not consider robustness, because robustness will come for watermarking definitely not for spatial domain steganography
Security analysis has been made to highlight its firmness against hackers
Total complexity is 264 *256!* 4 *4!*8/k
The work tested for 10 cover images, due to large data values, only four frequently used cover images are given in the result & discussion. Table 3 highlights the superiority of the proposed method with available literature
Usage of nominal key length reduces the cost associated with the transport of key over a secure channel about 0.098% of the embedded text
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