A Note On the Bounds for the Generalized Fibonacci-p-Sequence and its Application in Data-Hiding

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1 International Journal of Comuter Science and Alications c Technomathematics Research Foundation Vol. 7 No. 4,. 1-15, 010 A Note On the Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding Sandian Dey Microsoft India Develoment Center sandian.dey@gmail.com Hameed Al-Qaheri Deartment of Quantitative Methods and Information Systems College of Business Administration Kuwait University alqaheri@cba.edu.w Suneeta Sane Comuter and Information Technology Deartment Veermata Jijabai Technological Institute Mumbai, Maharashtra , India sssane@vjti.org.in Sugata Sanyal School of Technology and Comuter Science Tata Institute of Fundamental Research Homi Bhabha Road, Mumbai , India sanyal@tifr.res.in In this aer, we suggest a lower and an uer bound for the Generalized Fibonacci-Sequence, for different values of. The Fibonacci--Sequence is a generalization of the Classical Fibonacci Sequence. We first show that the ratio of two consecutive terms in generalized Fibonacci sequence converges to a -degree olynomial and then use this result to rove the bounds for generalized Fibonacci- sequence, thereby generalizing the exonential bounds for classical Fibonacci Sequence. Then we show how these results can be used to rove efficiency for data hiding techniques using generalized Fibonacci sequence. These steganograhic techniques use generalized Fibonacci--Sequence for increasing the number of available bit-lanes to hide data, so that more and more data can be hidden into the higher bit-lanes of any ixel without causing much distortion of the cover image. This bound can be used as a theoretical roof for efficiency of those techniques, for instance it exlains why more and more data can be hidden into the higher bit-lanes of a ixel, without causing considerable decrease in PSNR. Keywords: Fibonacci-sequence, LSB Data-hiding, PSNR 1. Introduction Among many different data hiding techniques roosed to embed secret message within images, the LSB data hiding technique is one of the simlest methods for inserting data into digital signals in noise free environments, which merely embeds secret message-bits in a subset of the LSB lanes of the image. LSB is the least 1

2 Dey, Al-Qaheri, Sane, Sanyal significant bit or the 0th bit, the second LSB is the 1st bit, and so on. Desite being simle, this technique is more redictable and hence less secure, also PSNR ea signal to noise ratio decreases very raidly as we use the higher bit lanes for data hiding. As soon as we go from LSB least significant bit to MSB most significant bit for selection of bit-lanes for our message embedding, the distortion in stego-image is liely to increase exonentially, so it becomes imossible without noticeable distortion and with exonentially increasing distance from cover-image and stego-image to use higher bit-lanes for embedding without any further rocessing. The worarounds may be: through the random LSB relacement in stead of sequential, secret messages can be randomly scattered in stego-images, so the security can be imroved. Also, using the aroaches given by variable deth LSB algorithm [Liu et al. 004], or by the otimal substitution rocess based on genetic algorithm and local ixel adjustment [Wang et al. 001], one is able to hide data to some extent in higher bit-lanes as well. Battisti et al. [Battisti et al. 006], [Picione et al. 006] roosed a novel data hiding technique from a totally different ersective, it uses a different bitlanes decomosition altogether, based on the generalized Fibonacci--sequences, that not only increases the number of embeddable bit-lanes but also decreases PSNR in the stego image considerably, thereby imroving the LSB technique. In this aer, we first rove some theoretical uer and lower bounds for generalized Fibonacci--sequence and give a theoretical roof for the better erformance of the data hiding technique using generalized Fibonacci decomosition, i.e., why the data hiding technique using this decomosition not only gives larger number of bit lanes for hiding secret bits, but also gives a far better PSNR than that in classical LSB technique. The Generalized Fibonacci--Sequence [Horadam 1961], [Basin and Hoggatt 1963], [Hoggatt 197], [Atins and Geist 1987], [Hendel 1994], [Sun and Sun 199] is given by, F 0 = F 1 =... = F = 1, F n = F n 1 + F n 1, n + 1, n, N 1 For = 1, we have, F 0 = F 1 = 1, F n = F n 1 + F n 1, n, n N We get the classical Fibonacci sequence 1, 1,, 3, 5, 8,.... We already have some results for this Classical Fibonacci Sequence, e.g., we now the ratio of two consecutive terms in Fibonacci sequence converge to Golden Ratio, In this aer, we show that α n > F n > α n, n, N, where α is the ositive Root of

3 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 3 x x 1 = 0. The ratio of two consecutive terms in Fibonacci--Sequence converges to this α.. Bounds for the generalized Fibonacci--Sequence In this section we rove the uer and lower bounds for the generalized Fibonacci-sequence. First, we show that the ratio of consecutive terms of generalized Fibonacci--sequence converges to the ositive root of the olynomial x x 1 = 0. Next we use this root to rove a bound on the generalized Fibonacci--sequence..1. Lemma 1 The ratio of two consecutive numbers in generalized Fibonacci -sequence converges to the ositive root of the degree- olynomial P x = x x 1. Proof: Convergence Let us first define{ the ratio} of two consecutive terms of Fibonacci--sequence as a Fn+1 sequence {β n } = F n. Now, by definition of Fibonacci--sequence, we have β 0 = β 1 = = β = 1 β n = 1 + F n > 1, n > F n β = β + β 0 = = β n = 1 + F n < =, n > + 1 F n 1 < β n <, n > + 1 We observe that the sequence β n is bounded and hence by Monotone Convergence theorem must have a convergent subsequence. For instance, for = 1 classical Fibonacci sequence, we have two convergent subsequences β n increasing and β n+1 decreasing, n N natural number and they both converge to the same limit [Craw 000], as shown in figure. Positive Root of the olynomial x x 1 Now, let s analyze the olynomial function y = P x = x x 1. By Descartes rule, the olynomial can have at most one ositive real root, since it has exactly one change in sign. First we observe that the function P x is continuous and differentiable everywhere. We also notice that the function has exactly one ositive root α and α

4 4 Dey, Al-Qaheri, Sane, Sanyal is also strictly larger than 1. This is a consequence of elementary calculus. By successive differentiation, we see that y 1 = P x = + 1x x 1 y = P x = + 1x 1 1x The function P x has critical oints at P x = 0, i.e., at x = 0 and x = y x=. = 1 > 0, 1. By nd order sufficient condition for local minima, P x has a local minima at x =. At x = 0, the function will have a maxima or a oint of inflection deending on whether is odd or even, exlained in the next section. When N odd, we have, y 1 = P x = + 1x 1 x is < 0 x < 0 = 0 x = 0 < 0 0 < x < = 0 x = > 0 x >, and increasing in Hence the function is decreasing in., Also, y 1 = 1, y0 = y1 = 1 and y = 1 1, N odd. At x =, P x has a minima gradient changes from negative to ositive but at x = 0 no sign change in gradient we have a oint of inflection. Again, P x being a continuous function assumes all ossible values within an interval. Combining all these, we can easily see that the grah of the function has exactly two real zeroes, one ositive and the other negative the remaining roots are comlex conjugate airs. When N even, we have, y 1 = P x = + 1x 1 x is > 0 x < 0 = 0 x = 0 < 0 0 < x < = 0 x = > 0 x > Hence the function is increasing in, 0, then decreasing in 0, and again increasing in., Also, y 1 = y0 = y1 = 1 and y = 1 1, N even. At x 0 =, P x has a minima but at x = 0 gradient changes from ositive to negative we have a maxima. Combining all these, we can easily see that we have exactly one real ositive root at α, since y1 < 0 and y > 0, also yx

5 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 5 being continuous. From this result we immediately have Lemma. From figure 1 we can see the grah of the degree- olynomial, for odd and even resectively. Convergence to the ositive root of the olynomial Now, we show that if sequence {β n } converges to β, then β = α. Let s assume the sequence {β n } converges to β R +. Now we rove, the sequence must converge to the only ositive root α of the above-stated -degree olynomial. By assumtion, β = lim n fn+ f n+ 1 = lim n fn+ 1 f n fn =... = lim =..., n f n 1 f n = n th number in the F ibonacci Sequence, f n+ = f n+ 1 + f n 1 fn+ 1 + f n 1 fn β = lim = lim, n n β = 1 + lim n f n+ 1 f =n+ =n 1 β = 1 + f +1 = =1 1 β = lim n f n 1 fn f n 1 β = β β β 1 = 0 Hence β satisfies the equation x x 1 = 0, N, β R +, i.e., from above results, we have, β = α Fig. 1. Grah of x x 1 = 0, showing α for different values of

6 6 Dey, Al-Qaheri, Sane, Sanyal.. Lemma If α be a ositive root of the equation x x 1 = 0, we have 1 < α <, N. Proof: We have, Also, α α 1 = 0 1 = 1 > 0, N 1 > α α 1 α > α α Also, 1 < 0 = α α 1 α α 1 > 0 From, we immediately see the following: α > 1 since ositive 3 α > 0 according to our assumtion, hence we can not have α = LHS & RHS both becomes 0, that does not satisfy inequality. If α >, we have LHS < 0 while RHS > 0 which again does not satisfy inequality. Hence we have α <, N From 3, we have, α > 1. Combining, we get, 1 < α <, N.3. Lemma 3 If α be a ositive root of the equation x x 1 = 0, where N, we have the following results, N, α > α +1 lim α = 1 α +1 > 1+α α < + 1 α +1 > 1 Proof: We have, For =, α +1 α 1 = 0 For = + 1, α + +1 α = 0 α α +1 1 = αα 1 α+1 1 = α 1 α α +1.α +1 4

7 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 7 From 4 we can argue, α α +1, since neither of them is 0 or 1 from Lemma. If α < α +1, we have LHS of inequality 4 < 1, but RHS > 1, since both the terms in RHS will be greater than 1 by our assumtion and by Lemma, a contradiction. Hence we must have Also, from Lemma, we have, 1 < α <, N. Hence we have, Again, from 4 we have, α > α +1, N 5 > α 1 > α >... > α > α +1 >... > 1, N lim α = 1 α+1 1 α.α +1 > 1, since > 1, from 5 α 1 α +1 α 1 > α +1 >, from Lemma α +1 1 α +1 > 1 + α Now, let us induct on N to rove α < + 1. Base case: For = 1, 1 < α 1 <, by Lemma. Let us assume the inequality holds < α < + 1, Induction Ste: = + 1, α = α 1 α., by 4 α +1 1 α < + 1. α 1 α +1 1 < α < + 1, by induction hyothesis α < α α +1 α +1 1 α < α α +1 α +1 1 α < , from 6, we have, α α +1 α +1 1 < 1 6 α < + α < + 1, N 7

8 8 Dey, Al-Qaheri, Sane, Sanyal Also, since α is a root of x x 1 = 0, for = we have. α +1 α 1 = 0 α +1 = α + 1 > since from Lemma, we have, α > 1 α +1 > 8.4. Lemma 4 The following inequalities always hold: < <... < 4 3 < 3 1 < α < + 1 α 1 α + > α +3 Proof: By Binomial Theorem, we have, = 1 r=0 1 1 <... α 3 < 4 α < 3 α < > 3... α + > α + > = 1 r! 1 r= r 1... r r!. 1 = 1 r=0. 1 r r s=1 1 s. 1 r! < =. 1 = < 1 9 }{{} times Hence we have, < 1 1 <... < < 3 1 < Also, from 7 we have, α < Combining, we get, α < < 1 1 <... < < 3 1 < α < + 1 α 1 <... α 4 < 5 α 3 < 4 α < 3 α < 10 Also, we have, α + > α +3 α +1 > α + = α +1 + α > + 1 = 3 α + > 3 α +3 = α + + α > = 4 > 3... α + > α + > Lemma 5 The following inequality gives us the lower and uer bounds for generalized Fibonacci--sequence, α n > F n > α n, n >, n N 1 where α is the ositive root of the equation x x 1 = 0.

9 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 9 Proof: We induct on n to show the result. F 0 = F 1 =... = F = 1, By definition of Fibonacci--Sequence. Base case: From Lemma 4, we have, α α + α +3 > F + 1 = F + F 0 = = > α > F + = F F 1 = + 1 = 3 > α > F + 3 = F + + F = = 4 > α 3 α + > F = F + + F = = + > α Induction Ste: Let s assume the above result is also true m : + 1 < m < n, m, n N. Now, we rove for m = n, α n 1 + α n 1 F n 1 + F n 1 > α n 1 Hence we have the following inequality, > F n 1 + F n 1 by hyothesis + α n 1 by hyothesis α n α > F n > α n α α n 1.α > F n > α n 1.α α n > F n > α n, n >, n N α n > F n > α n, α R + and α 1, 13 α 1 = , α , α , α , α > α, N The emirical results Table 1 also rove our claim for =. Also F 0 = F 1 = = F = 1 and F n + 1 > F n, n >. Hence we have, F n = F n 1 + F n 1 <.F n 1, n > F n <.F n 1 <.F n < < n.f = n, n > F n < n, n > 14

10 10 Dey, Al-Qaheri, Sane, Sanyal Combining 13 and 14, we have, n > F n > α n, n >, and n, N 15 where α is the ositive Root of x x 1 = 0. Figure and Table show the convergence of ratio of successive terms for Fibonacci--sequences for different For = 1 we get classical Fibonacci sequence. It can be noticed that smaller the value of, quicer the convergence of the ratio is achieved, as shown. Also value to which the ratio converges monotonically decreases with increase in the value of. 3. Alication in Data Hiding Data hiding is a new ind of secret communication technology, where message is hidden inside an image or any other medium, so that it cannot be observed. One of the simlest data hiding technique is LSB data hiding technique, which merely embeds secret message-bits in a subset of the LSB lanes of the image. One of the drawbacs of this technique is: as soon as we go from LSB to MSB for selection of bit-lanes for our message embedding, the distortion in stego-image is liely to increase exonentially, so it becomes imossible without noticeable distortion and with exonentially increasing distance from cover-image and stego-image to use higher bit-lanes for embedding without any further rocessing. This articular roblem was addressed by Battisti et al., [Battisti et al. 006], who roosed to use generalized Fibonacci--sequence decomosition technique instead of classical binary decomosition and shows by emirical results that this

11 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 11 Fig.. a Convergence of the ratio of successive terms in generalized Fibonacci -Sequence for different values of b Convergent Subsequences for classical Fibonacci sequence =1 technique outerforms the classical LSB technique when thought in terms of embedding in the higher bit lane as well with less distortion. This technique basically increases the number of bit-lanes by generating a new larger set of bit-lanes that we call virtual bit-lanes by using Fibonacci--sequence decomosition. It can be further imroved using rime and natural number decomosition techniques

12 1 Dey, Al-Qaheri, Sane, Sanyal as shown in [Dey et al. 007a], [Dey et al. 007b], [Dey et al. 008]. Also, [Cooer 1984] [Dotson et al. 1993] illustrates Fibonacci sequence can be used in various alications. In this aer we give a formal roof of Fibonacci--sequence bounds and show how this can be used to theoretically rove that Fibonacci--sequence decomosition gives better result in hiding data. From 15 it is clear from the uer bound that the same value will require more numbers of bits to be reresented than the number of bits required in classical binary decomosition since n > α n > F n, if it s exressed using Fibonacci--sequence decomosition where the radix is Fibonacci-sequence numbers instead of owers of. As illustrated in [Dey et al. 008], in order to measure the distortion in the stego-image, we use Mean square error MSE, Worst case Mean Square Error

13 Bounds for the Generalized Fibonacci--Sequence and its Alication in Data-Hiding 13 WMSE and Pea Signal to Noise Ratio PSNR, which are defined by MSE = M i=1 j=1 N f ij g ij /MN P SNR = 10.log 10 L MSE [Dey et al. 008]. If the secret data-bit is embedded in the i th bit-lane of a ixel, the worst-case error-square-er-ixel will be = W SE = W i1..0 = W i here W i reresents the corresonding weight in the number system for the i th bit, e.g., for classical decomosition W i = i, for generalized Fibonacci decomosition W i = F i, corresonding to the case when the corresonding bit in cover-image toggles in stego-image, after embedding the secret data-bit. For examle, worst-case error-square-er-ixel for embedding a secret data-bit in the ith bit lane in case of a ixel in classical binary decomosition is = i = 4 i, where i N {0}. If the original -bit grayscale cover-image has size w h, we define, W MSE = w h W i = w h W SE [Dey et al. 008]. Hence, WMSE after embedding secret message bit only in the l th virtual bit-lane of each ixel in case of classical traditional binary LSB data hiding technique is given by, W MSEl th bit lane Classical Binary Decomosition = θ4l. WMSE after embedding secret message bit in the l th virtual bit-lane of each ixel in case of generalized Fibonacci decomosition is given by, W MSEl th bit lanef ibonacci Sequence Decomosition = F l α l < W MSEl F ibonacci sequence < α l, α R +, α 1 = 1 + 5, α > α, N, α1.618, W MSE l Generalized F ibonacci sequence < θ.618 l. Hence, we have, W MSE Binary > W MSE F ibonacci P SNR F ibonacci > P SNR Binary. Thus, we have first roved bounds on the generalized Fibonacci sequence and then by using our bounds, we have given a formal roof for better erformance in terms of PSNR for LSB data hiding technique using generalized Fibonacci-sequence decomosition than that using classical binary decomosition. Also, we

14 14 Dey, Al-Qaheri, Sane, Sanyal have, number of rimes n = n = θn. log n = oα n < F n < n [Telang 1999], [Niven and Zucerman 1966], [Tattersall 005] The above imlies that if the same number is reresented using rime decomosition n th rime number as weightage to n th bit, it will give still more numbers of virtual bit lanes [Battisti et al. 006]. We have similar results for LSB data hiding using natural number decomosition technique, and combining the results from [Battisti et al. 006], [Dey et al. 007a], [Dey et al. 007b], [Dey et al. 008] we have the following, W MSE Binary > W MSE F ibonacci > W MSE P rime > W MSE Natural P SNR Natural > P SNR P rime > P SNR F ibonacci > P SNR Binary. Hence, data hiding using natural number decomosition gives the best erformance among the above mentioned techniques. In data hiding, we hide data in different bit-lanes of a ixel. In classical LSB data-hiding technique the ixel is reresented as binary value, hence it has less numbers of bit-lanes as we have in case of Fibonacci--sequence decomosition, the later having still less number of bit-lanes than in case of rime decomosition technique. It is shown in [Dey et al. 007a], [Dey et al. 007b], [Dey et al. 008] by calculation of WMSE and PSNR measures that embedding data even in higher bit-lanes of ixel using these techniques results in less visible distortion of the cover image, since the distortion as measured by WMSE is roortional to the square of the weights to the bits in the corresonding decomosition, hence it decreases as the weights go on decreasing from classical binary to generalized Fibonacci and from that to rime and natural number decomosition [Dey et al. 008]. 4. Conclusions In this aer, we have established the bounds for generalized Fibonacci-sequence α n > F n > α n, n >, n, N. Emirical results obtained vindicates our theoretically-roven bounds. Then we used the result n > F n > α n, n >, n, N, where α is the ositive Root of x x 1 = 0, to rove that data hiding technique using generalized Fibonacci--sequence gives more embeddable bit-lanes along with better PSNR than that in case of classical LSB technique [Dey et al. 008], and the same using rime decomosition technique increases virtual bit-lanes and PSNR further [Dey et al. 007a]. References Battisti F., Carli M., Neri A., Egiazarian K A Generalized Fibonacci LSB Data Hiding Technique. 3rd International Conference on Comuters and Devices for Communication CODEC- 06, December 18 0 TEA, Institute of Radio Physics and Electronics, University of Calcutta.

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