Techniques in Image Steganography using Famous Number Sequences

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INTRODUCTION
The exchange of information is essential for the development of civilization. The discovery and evolutionof methods that could make transmission of information secure attracted people since antiquity. Over thecenturies people discovered and developed techniques which evolved into the sciences of Crytography andSteganography. Cryptography disguises the message to be trasmitted so that only the intended recipient isable to read it, while steganography hides the message by embedding it within other, seemingly harmless,messages. Steganography dates back to the ancient Greece but only lately (late 20th century) it began beingresearched for scientific reasons. Today, it is widely used in Telecommunications, Industry [1] , Medicine [2]and in the practice of hiding strongly encrypted data. Steganography, in contrast to Cryptography, is not trying to make a message incomprehensible for an invalid person [3], [4], but to hide its existence, using acover, e.g., by incorporating the message to be transmitted into an image.

The LSB method
In recent years, the LSB (Least Significant Bit) method became one of the most important steganographic methods for hiding data within images [5]. In a × color RGB image, with 8 bit color depth, each pixel assumes an integer value x on the closed interval [0, 255] for each color (Red, Green, Blue). The number represents the density of the color and it is encoded by an 8 bit binary word 7  This definition of allows the decomposition of an image into a collection of binary images by separatingthe bits into 8 bit planes. In the classical LSB embedding methods, the secret message is inserted into theleast significant bitplane, i.e., the 8th bitplane, of the cover image, either by directly replacing those bits orby modifying them using a particular "inverse" function [6] ( Figure 1).The embedding strategy can also be based on sequential insertion or selective embedding of the messagein "noisy" areas or random scattering throughout the image [7]. The Fibonacci method Several research teams have developed and extended the LSB method using different approaches. One ofthese approaches is presented in [9], and uses the Fibonacci numbers, which are defined by the linearrecurrence relation F n = F n−1 + F n−2 , n > 1, with F 0 = 0 and F 1 = 1.
These three sums can be represented respectively by the binary words: 000010000110 12 , 000010001000 12 , 000001101000 12 . Therefore, the number 39 has more than one representations, in terms of Fibonacci numbers. Fromall these representations, we choose the one which is derived from Zeckendorf's Theorem, i.e., the onecorresponding to the binary word 000010001000 12 , (which is the only one containing no consecutive 1's). Inthis way, we produce 12 bitplanes for embedding data and so we can increase the stego capacity.

An extension of Zeckendorf's Theorem
In this section, we give an extension of Zeckendorf's theorem, which allows us to use other integer sequencesfor the representation of bytes. This way, we improve the previously described Fibonacci method. Some oftheses results were presented in [10].
Theorem 1 (Extension of Zeckendorf's Theorem) . Let (a n ) n∈ℕ * be a strictly increasing sequence of positiveintegers, with a 1 = 1, a 2 = 2, and a n + a n+1 ≥ a n+2 and n ∈ ℕ * . Then, every positive integer x witha n ≤ x < +a n+1 , n ∈ ℕ * can be represented as a sum of different and nonconsecutive terms of the sequence(a n ), with the restriction that the term a n appears in the sum.
Proof. We will use induction on x. If 1 ≤ x ≤ 3, then the claim obviously holds, since a 1 = 1, a 2 = 2,a 3 = 3. Let x > 3, and assume that the claim holds for all positive integers less than x. Then, there existsa unique n > 2, such that a n ≤ x < +a n+1 . If a n = x, then the claim obviously holds. If a n < < +a n+1 ,then, setting y = x − a n , by the induction hypothesis, we have that y is represented as a sum of differentand nonconsecutive terms of the (a n )sequence. Lety = a b λ + a b λ −1 + ⋯ + a b 1 be a represention of y , where (b n ) is a strictly increasing sequence of nonconsecutive positive integers, andλ ∈ ℕ * . Therefore, given the restriction that the term a n must appear in the representation of x , we obtainthe representation x = a n + y of x . Furthermore, we have that < a n+1 ⇒ y < a n+1 − a n and sincea n+1 − a n ≤ a n−1 , it follows that y < a n−1 . The representation of y gives that a b λ ≤ y, thusa b λ < a n−1 . Finally, since the sequence (a n )is strictly increasing, we havea b λ < a n−1 ⇒ b λ < − 1,so that the representation of x contains no consecutive terms.
According to Theorem 1, given a sequence (a n )satisfying the above requirements, any x ∈ ℕ is representedas where w i ∈ 0,1 , w n = 1and there is no, such that w i = w i+1 = 1. The number n is theunique positive integer satisfying a n < < +a n+1 . Therefore, each representation corresponds to a uniqueFibonacci word w n w n−1 … w 1 , so that each x ∈ ℕcorresponds to at least one Fibonacci word. By choosingthe lexicographically greatest corresponding word, we define an encoding for the elements of ℕ. This isequivalent to applying recursively the restriction of the Theorem. The implementation for this is trivial and,therefore, the process of encoding and decoding each integer x is straightforward.
These representations correspond to the Fibonacci wordsw = 1010000 and u = 1001010respectively. Since w is greater than u , the number 18 is encoded by w .
Corollary 2. Let (a n ) n∈ℕ * be a strictly increasing sequence of positive integers, such that the Fibonaccinumbers F 2 , F 3 , F 4 , … form a subsequence of (a n ). Then, every positive integer x, with a n < < +a n+1 ,n ∈ ℕ * can be represented as a sum of different and nonconsecutive terms of the sequence (a n ), with therestriction that the term an appears in the sum. O c t 1 5 , 2 0 1 3 Proof. By the definition of (a n ), it follows that a 1 = F 2 = 1, a 2 = F 3 = 2and a 3 = F 4 = 3. Hence, byTheorem 1, it suffices to prove that a n + a n+1 ≥ a n+2 . This is obviously true for n = 1. If n ≥ 2, then bythe definition of (a n ), it follows that there exists a unique k ∈ ℕ * , where 2 ≤ k ≤ n, such that F k ≤ a n < a n+1 ≤ F k+1 If, a n+1 ≤ F k+1 then a n+2 ≤ F k+1 so that F k ≤ a n < a n+1 < F k+1 ⇒ a n + a n+1 > F k−1 + F k = F k+1 ≥ a n+2 On the other hand, if a n+1 = F k+1 , then a n+2 ≤ F k+2 , so that F k ≤ a n < a n+1 = F k+1 ⇒ a n + a n+1 > F k + F k+1 = F k+2 ≥ a n+2 Thus, in both cases we have that + +1 > +2
The above two sums are encoded respectively by the binary words:000010010100 12 , 000010010011 12 .
As in the case of Fibonacci numbers, we use the representation 000010010100 12 which is the lexicographicallygreatest.
The method using Catalan numbers. We use the sets and for the representation of the color values of each pixel. Using the set CF forthe representation of the color values of each pixel, we create 15 virtual bitplanes, 3 more than the numberof bitplanes produced by the Fibonacci method. Using the set , we create 16 virtual bitplanes, 4 more than the Fibonacci and the Lucas representation. In this way, more stego data can be embedded into theimage.
The procedure for this method is as follows. Firstly, each pixel value is represented by its decimal value.Then, this value is converted using (or ) numbers, to 15 (or 16) bitplanes. So, the message can beembedded in the last bitplane (as in LSB) as well as in higher bitplanes.

Measures and results
We examine the effectiveness of our methods, by comparing the quality of the correspoding (resulting)images. The implementation of our method is done using our own application (Crypto ver. 1.2), which ispowered by MATLAB software. As test image, we use the three grayscale images: baboon, airplane andpepper. (Figure 2). O c t 1 5 , 2 0 1 3

Fig 2: Test Images
We create four stego images, each one using a different method (LSB, Fibonacci, Lucas, Catalan-Fibonacci, Catalan-Lucas method).
We use two metrics to compare the various image techniques: The Mean Square Error (MSE) and thePeak Signal to Noise Ratio (PSNR). A lower value for MSE means lower error, and as seen from the inverserelation between MSE and PSNR, this translates to a high value of PSNR. MSE is the cumulative squarederror between the stego image and the original image and is defined as follows: where and are two × monochrome images, where one of the images is considered to be a noisyapproximation of the other. PSNR is given by the formula: where is the maximum possible pixel value of the image. When the pixels are represented using 8bits per sample the value of is 255.
We present the measures of PSNR, when embedding data in more bitplanes, in tables 1, 2, 3, 4, 5, 6 and7.For pages other than the first page, start at the top of the page, and continue in single-column format.
In the next tables and in columns 2, 3, 4, 5 and 6, we see PSNR measurements for each method. We cansee that our method improves the performance of the image quality, when compared to the simple LSB andthe Fibonacci methods. Moreover, the Lucas sequence and the sets CF and CL give better results than theLSB and Fibonacci methods, when we embed data in higher bitplanes. Of all the methods, the one usingthe set CL seems to give better results. More specifically, the PSNR value in the Lucas method and in theCatalan-Lucas method is increased by about 2.5% and 13% respectively, compared to the Fibonacci method,while the Catalan-Lucas method improves the PSNR value of the Catalan-Fibonacci method by about 5%.In figure 5, we can see an average PSNR comparison and in figure 6, we see what happens to the picture when we overflow the image with stego data   In figure 3, we can see an average PSNR comparison and in figure 4, we see what happens to the picture when we overflow the image with stego data proposed an improvement over this by using Fibonacciembedding technique and our method, using a greater set of virtual bitplanes, increases the number of stegobits that can be embedded when an image should be regarded as a stego image. Besides, apart from theLSB method, in which the tracking of data is easy (if there is suspicion of course), the Fibonacci, Lucas, CFand CL methods offer a kind of encryption, through the way that they conceal the bits in more bitplanes. Particularly, our method enables a large number of bitplanes, offering not only more space for our stego databut also increased security towards steganalysis software.