Sandipan Dey (1), Ajith Abraham (2), Sugata Sanyal (3) 1Anshin Software Private Limited, Kolkata – 700091 2Centre for Quantifiable Quality of Service in Communication Systems Norwegian University of Science and Technology, Norway 3School of Technology and Computer Science, Tata Institute of Fundamental Research, India

sandipan.dey@gmail.com, ajith.abraham@ieee.org, sanyal@tifr.res.in

Abstract In this paper, a novel data hiding technique is proposed, as an improvement over the Fibonacci LSB data-hiding technique proposed by Battisti et al. [1] based on decomposition of a number (pixel-value) in sum of natural numbers. This particular representation again generates a different set of (virtual) bit-planes altogether, suitable for embedding purposes. We get more bitplanes than that we get using Prime technique [2]. These bitplanes not only allow one to embed secret message in higher bit- planes but also do it without much distortion, with a much better stego-image quality, and in a reliable and secured manner, guaranteeing efficient retrieval of secret message. A comparative performance study between the classical Least Significant Bit (LSB) method, the Fibonacci LSB data-hiding technique and the proposed schemes indicate that image quality of the stego-image hidden by the technique using the natural decomposition method improves drastically against that using Prime and Fibonacci decomposition technique. Experimental results also illustrate that, the stego-image is visually indistinguishable from the original cover-image. Also we show the optimality of our technique.

1. Introduction Data hiding technique is a new kind of secret communication technology. While cryptography scrambles the message so that it can’t be understood, steganography hides the data so that it can’t be observed. In this paper, we discuss about a new decomposition method for classical LSB data-hiding technique, in order to make the technique more secure and hence less predictable. We generate a new set of (virtual) bit planes using our decomposition technique and embed data bit in these bit planes. The Fibonacci LSB Data Hiding Technique proposed by Battisti et al. [1] investigates decomposition into a different set of bit-planes, based on the Fibonacci– p-sequences, given by, Ν ∈ ≥ ∀ − −

−

=

n n p n F n F n F F F p p p p p ,2 ), 1 ( )1 ( ) ( 1 )1( ) 0 (

and embed a secret message-bit into a pixel if it passes the Zeckendorf condition, then during extraction, follow the reverse procedure.
We proposed the data hiding technique using prime decomposition [2] as an improvement over Fibonacci. Virtual bit-planes are generated using Prime Decomposition. The weight function of the Prime Number System is defined as: ,.. 5 ,3 ,2 ,1 , Pr , , ) ( ,1 ) 0 ( 3 2 1 0

=

=

∈ ∀

= + p p p p ime i p Z i p i P P th i i

and embed a secret message-bit into a pixel if after embedding it still remains as a valid representation. It has been shown that this technique not only increases the options for embedding by increasing number of bit-planes but also gives less distortion than classical binary and Fibonacci Decomposition, while embedding message in higher bit-planes [2]. Rest of the paper is organized as follows. In Section 2, the proposed natural decomposition technique is introduced followed by experiment results in Section 4. Some conclusions are also provided towards the end. 2. Natural Number Decomposition

We define another new number system denoted as (.)) ,2 ( Ν , where the weight function (.) Ν is defined as: { } 0 ,1 ) ( ) ( ∪ ∈ ∀ +

Ν

Z i i i i W

Since the weight function is composed of natural numbers, we name this number system as natural number system and the decomposition as natural number decomposition. In this number system, we have redundancy too. To make our transformation one-to-one, we again take the lexicographically highest of all the presentations in our number system, corresponding to same value. (e.g., value ‘3’ has two different representations in 3-bit natural number system, namely, 100 and 011, since
3 1.1 2.1 3.0, ,3 1.0 2.0 3.1

= + + and

Since 100 is lexicographically (from left to right) higher than 011, we choose 100 to be valid representation for 3 in our natural number system and thus discard 011, no longer a valid representation in our number system. 100 ) 011 , 100 ( max 3 ≡ ≡ ic lexicogaph

In our example, the valid representations are: 6 111 ,5 110 ,4 101

3, 100

2, 010

1, 001

0, 000 ↔ ↔ ↔ ↔ ↔ ↔ ↔

Also, to avoid loss of message, we embed secret data bit to only those pixels, where, after embedding we get a valid representation in the number system. It’s worth noticing that, up-to 3-bits, the prime [2] and the natural number system are identical, after that they are different. 2.2. Embedding

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