A new chaos-based watermarking algorithm
aa r X i v : . [ c s . MM ] O c t A new chaos-based watermarking algorithm
Christophe Guyeux
Computer Science Laboratory (LIFC)University of Franche-Comt´eRue Engel-Gros, BP 527, 90016 Belfort Cedex, [email protected]
Jacques M. Bahi
Computer Science Laboratory (LIFC)University of Franche-Comt´eRue Engel-Gros, BP 527, 90016 Belfort Cedex, [email protected]
Abstract —This paper introduces a new watermarking algo-rithm based on discrete chaotic iterations. After defining somecoefficients deduced from the description of the carrier medium,chaotic discrete iterations are used to mix the watermark andto embed it in the carrier medium. It can be proved that thisprocedure generates topological chaos, which ensures that desiredproperties of a watermarking algorithm are satisfied.
I. INTRODUCTIONInformation hiding has recently become a major securitytechnology, especially with the increasing importance andwidespread distribution of digital media through the Internet.It includes several techniques, among which is digital water-marking. The aim of digital watermarking is to embed a pieceof information into digital documents, like pictures or moviesfor example. This is for a large panel of reasons, such as: copy-right protection, control utilization, data description, integritychecking, or content authentication. Digital watermarking musthave essential characteristics including imperceptibility androbustness against attacks. Many watermarking schemes havebeen proposed in recent years, which can be classified into twocategories: spatial domain (Wu et al., 2007) and frequency do-main watermarking (Cong et al., 2006), (Dawei et al., 2004).In spatial domain watermarking, a great number of bits can beembedded without inducing too clearly visible artifacts, whilefrequency domain watermarking has been shown to be quiterobust against JPEG compression, filtering, noise pollution,and so on. More recently, chaotic methods have been proposedto encrypt the watermark, or embed it into the carrier imagefor security reasons.In this paper, a new watermarking algorithm is given. It isbased on the commonly named chaotic iterations and on thechoice of relevant coefficients deduced from the description ofthe carrier medium. This new algorithm consists of two basicstages: a mixture stage and an embedding stage. At each ofthese two stages, the proposed algorithm offers additional stepsthat allow the authentication of relevant information carriedby the medium or the extraction of the watermark withoutknowledge about the original image.This paper is organized as follows: firstly, some basicdefinitions concerning chaotic iterations is recalled. Then,the new chaos-based watermarking algorithm is introduced inSection III. Section IV is constituted by the evaluation of ouralgorithm: a case study is presented, some classical attacks areexecuted and the results are presented and commented on. The paper ends by a conclusion section where our contribution issummarized, and planned future work is discussed.II. BASIC RECALLS: CHAOTIC ITERATIONSIn the sequel S n denotes the n th term of a sequence S , V i denotes the i th component of a vector V and f k = f ◦ ... ◦ f denotes the k th composition of a function f . Finally, thefollowing notation is used: J N K = { , , . . . , N } .Let us consider a system of a finite number N of cells , sothat each cell has a boolean state . Then a sequence of length N of boolean states of the cells corresponds to a particular stateof the system . A sequence which elements belong in J N K iscalled a strategy . The set of all strategies is denoted by S . Definition 1
Let S ∈ S . The shift function is defined by s : ( S n ) n ∈ N ∈ S −→ ( S n + ) n ∈ N ∈ S and the initial function i is themap which associates to a sequence, its first term: i : ( S n ) n ∈ N ∈ S −→ S ∈ J N K . Definition 2
The set B denoting { , } , let f : B N −→ B N bea function and S ∈ S be a strategy. Then, the so-called chaoticiterations are defined by x ∈ B N and ∀ n ∈ N ∗ , ∀ i ∈ J N K , x ni = (cid:26) x n − i if S n = i (cid:0) f ( x n − ) (cid:1) S n if S n = i . (1)III. A NEW CHAOS-BASED WATERMARKINGALGORITHM A. Most and Least Significant Coefficients
Let us first introduce the definitions of most and least signifi-cant coefficients of an image.
Definition 3
For a given image, the most significant co-efficients (in short MSCs), are coefficients that allow thedescription of the relevant part of the image, i.e. its most richpart (in terms of embedding information), through a sequenceof bits.For example, in a spatial description of a grayscale image,a definition of MSCs can be the sequence constituted by thefirst three bits of each pixel.
Definition 4
By least significant coefficients (LSCs), wemean a translation of some insignificant parts of a medium ina sequence of bits (insignificant can be understand as: “whichcan be altered without sensitive damages”).he LSCs are used during the embedding stage: some ofthe least significant coefficients of the carrier image will bechaotically chosen and replaced by the bits of the (possiblymixed) watermark.The MSCs are only useful in case of authentication, mixtureand embedding stages will then depend on them. Hence, acoefficient should not be defined at the same time both as aMSC and a LSC: the LSC can be altered, while the MSC isneeded to extract the watermark (in case of authentication).
B. Stages of the Algorithm
Our watermarking scheme consists of two classical stages:the mixture of the watermark and its embedding into a coverimage.
1) Watermark mixture:
For security reasons, the watermarkcan be mixed before its embedding. A common way toachieve this stage is to use the bitwise exclusive or (XOR),for example, between the watermark and a logistic map.In this paper, we will introduce a mixture scheme basedon chaotic iterations. Its chaotic strategy will be highlysensitive to the MSCs, in case of an authenticated water-mark (Bahi and Guyeux, 2010). For the details of this stagesee the Paragraph IV-A2 in Section IV.
2) Watermark Embedding:
This stage can be done either byapplying the logical negation of some LSCs, or by replacingthem by the bits of the possibly mixed watermark.To choose the sequence of LSCs to be changed, a numberof integers, less than or equals to the number N of LSCs,corresponding to a chaotic sequence (cid:0) U k (cid:1) k , is generated fromthe chaotic strategy used in the mixture stage and possibly thewatermark. Thus, the U k − th least significant coefficient of thecarrier image is either switched, or substituted by the k th bit ofthe possibly mixed watermark. In case of authentication, sucha procedure leads to a choice of the LSCs which are highlydependent on the MSCs.On the one hand, when the switch is chosen, the water-marked image is obtained from the original image, whoseLSCs L = B N are replaced by the result of some chaotic itera-tions. Here, the iterate function is the vectorial boolean nega-tion, defined by f : B N −→ B N , ( x , . . . , x N ) ( x , . . . , x N ) ,the initial state is L and strategy is equal to (cid:0) U k (cid:1) k . In this case,it is possible to prove that the whole embedding stage satisfiestopological chaos properties (Bahi and Guyeux, 2010), but theoriginal medium is needed to extract the watermark.On the other hand, when the selected LSCs are substi-tuted by the watermark, its extraction can be done with-out the original cover. In this case, the selection of LSCsstill remains chaotic, because of the use of a chaoticmap, but the whole process does not satisfy topologicalchaos (Bahi and Guyeux, 2010): the use of chaotic iterationsis reduced to the mixture of the watermark. See the Para-graph IV-A3 in Section IV for more details.
3) Extraction:
The chaotic sequence U k can be regener-ated, even in the case of an authenticated watermarking: theMSCs have not been changed during the stage of embeddingwatermark. Thus, the altered LSCs can be found. So, in case of substitution, the mixed watermark can be rebuilt and“decrypted”. In case of negation, the result of the previouschaotic iterations on the watermarked image, is the originalimage.If the watermarked image is attacked, then the MSCs willchange. Consequently, in case of authentication and due to thehigh sensitivity of the embedding sequence, the LSCs designedto receive the watermark will be completely different. Hence,the result of the decrypting stage of the extracted bits willhave no similarity with the original watermark.IV. A CASE STUDY A. Stages and Details1) Images Description:
Carrier image is the famous Lena,which is a 256 grayscale image and the watermark is the 64 ×
64 pixels binary image depicted in Fig. 1a. The embeddingdomain will be the spatial domain. The selected MSCs arethe four most significant bits of each pixel and the LSCs arethe three following bits (a given pixel will at most be modifiedby four levels of gray by an iteration). The last bit is then notused. Lastly, LSCs of Lena are substituted by the bits of themixed watermark. (a) Watermark. (b) Watermarked Lena.
Figure 1: Watermark and watermarked Lena.
2) Mixture of the Watermark:
The initial state x of thesystem is constituted by the watermark, considered as aboolean vector. The iteration function is the vectorial logicalnegation f and the chaotic strategy ( S k ) k ∈ N will depend onwhether an authenticated watermarking method is desiredor not, as follows. A chaotic boolean vector is generatedby a number T of iterations of a logistic map ( ( µ , U ) parameters will constitute the private key). Then, in case ofunauthenticated watermarking, the bits of the chaotic booleanvector are grouped six by six, to obtain a sequence of integerslower than 64, which will constitute the chaotic strategy. Incase of authentication, the bitwise exclusive or (XOR) ismade between the chaotic boolean vector and the MSCs andthe result is converted into a chaotic strategy by joining itsbits as above. Thus, the mixed watermark is the last booleanvector generated by the chaotic iterations.
3) Embedding of the Watermark:
To embed the watermark,the sequence ( U k ) k ∈ N of altered bits taken from the M LSCsmust be defined. To do so, the strategy ( S k ) k ∈ N of the mixture NAUTHENTICATION AUTHENTICATIONSize (pixels) Similarity Size (pixels) Similarity10 99.08% 10 89.81%50 97.31% 50 54.54%100 92.43% 100 52.24%
Table I: Zeroing attacks.
UNAUTHENTICATION AUTHENTICATIONAngle Similarity Angle Similarity5 94.67% 5 59.47%10 91.30% 10 54.51%25 80.85% 25 50.21%
Table II: Rotation attacks. stage is used as follows (cid:26) U = S U n + = S n + + × U n + n ( mod M ) . (2)To obtain the result depicted in Fig. 1b.Remark that the map q q of the torus, which is a famousexample of topological Devaney’s chaos (Devaney, 2003), hasbeen chosen to make ( U k ) k ∈ N highly sensitive to the chaoticstrategy. As a consequence, ( U k ) k ∈ N is highly sensitive tothe alteration of the MSCs: in case of authentication, anysignificant modification of the watermarked image will leadto a completely different extracted watermark. B. Simulation Results
To prove the efficiency and the robustness of the proposedalgorithm, some attacks are applied to our chaotic water-marked image. For each attack, a similarity percentage withthe watermark is computed, this percentage is the number ofequal bits between the original and the extracted watermark.
1) Zeroing Attack:
In this kind of attack, some pixels of theimage are put to 0. In this case, the results in Table I have beenobtained. We can conclude that in case of unauthentication, thewatermark still remains after a cropping attack: the desiredrobustness is reached. In case of authentication, even a smallchange of the carrier image leads to a very different extractedwatermark. In this case, any attempt to alter the carrier imagewill be signaled.
2) Rotation Attack:
Let r q be the rotation of angle q around the center ( , ) of the carrier image. So, thetransformation r − q ◦ r q is applied to the watermarked image.The good results in Table II are obtained.
3) JPEG Compression:
A JPEG compression is applied tothe watermarked image, depending on a compression level. Letus notice that this attack leads to a change of the representationdomain (from spatial to DCT domain). In this case, the resultsin Table III have been found. A good authentication throughJPEG attack is obtained. As for the unauthentication case, thewatermark still remains after a compression level equal to 10.This is a good result if we take into account the fact that weuse spatial embedding.
4) Gaussian Noise:
Watermarked image can be also at-tacked by the addition of a Gaussian noise, depending on astandard deviation. In this case, the results in Table IV havebeen found.
UNAUTHENTICATION AUTHENTICATIONRatio Similarity Ratio Similarity2 82.95% 2 54.39%5 65.23% 5 53.46%10 60.22% 10 50.14%
Table III: JPEG compression attacks.
UNAUTHENTICATION AUTHENTICATIONStandard dev. Similarity Standard dev. Similarity1 74.26% 1 52.05%2 63.33% 2 50.95%3 57.44% 3 49.65%
Table IV: Gaussian noise attacks.
V. DISCUSSION AND FUTURE WORKIn this paper, a new way to generate watermarking meth-ods is proposed. The new procedure depends on a generaldescription of the carrier medium to watermark, in termsof the significance of some coefficients we called MSC andLSC. Its mixture and also the selection of coefficients to alterare based on chaotic iterations, which generate topologicalchaos in the sense of Devaney. Thus, the proposed algorithmpossesses expected desirable properties for a watermarkingalgorithm. For example, strong authentication of the carrierimage, security, resistance to attacks, and discretion.The algorithm has been evaluated through attacks andthe results expected by our study have been experimentallyobtained. The aim was not to find the best watermarkingmethod generated by our general algorithm, but to give asimple illustrated example to prove its feasibility. In futurework, other choices of iteration functions and chaotic strategieswill be explored. They will be compared in order to increaseauthentication and resistance to attacks. Lastly, frequencydomain representations will be used to select the MSCs andLSCs. REFERENCESBahi, J. M. and Guyeux, C. (2010). Topological chaosand chaotic iterations, application to hash functions. In
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