Information Theoretical Analysis of Digital Watermarking. Multimedia Security

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Cuthbert Carr
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1 Information Theoretical Analysis of Digital Watermaring Multimedia Security

2 Definitions: X : the output of a source with alphabet X W : a message in a discrete alphabet W={1,2,,M} Assumption : X is a discrete alphabet, follows a discrete distribution S 0,1 : a rv. which indicates whether X will be watermared. P x The variable S is introduced in the model only to provide the possibility of expressing mathematically the existence or nonexistence of a watermar in a simple way. 2

3 K : a secret ey defined on a discrete alphabet. X S=1 : (watermared version) f 1 : x w y (The output of the watermaring function ) f S=0 : X 0 Y (non-watermared version) f 0 : 1 f Y x y f 1 The output of the watermaring function f 1 depends on the value of K, a secret ey which uniquely identifies the copyright owner. 3

4 S X W Y Z K fs P Z Y g ψ q Xˆ Wˆ Ŝ General model of a watermaring system 4

5 The watermared version Y then passes through a noisy channel and is transformed into Z y. This channel models both unintentional distortions suffered by Y and attacs aimed at deleting or corrupting the watermar information. In both cases we assume that the secret ey is not nown, so the noisy channel can be defined by the distribution P Z Y z y which is independent of K. 5

6 x X Finally, Z is processed to obtain a point will be used by the recipient instead of X. There are two tests that can serve to verify the ownership of Z : the watermar detection test q : y 0,1 the watermar decoding test : y w the detection test is used to obtain an estimate which of S (to decide whether Z has been watermared using ) the decoding test is used to obtain an estimate of W. s w 6

7 Imperceptibility : d : x x Let be a perceptually significant distortion. A watermaring system must guarantee that the functions f 0, f1 and g introduce imperceptible alternations with respect to X. E d x, g f x D 0 0 x, g f 1 x, w, D1 E d With expectations taen wrt. X, W, K, (Mean Distortion Consraints) 7

8 8 or (Maximum constraints) K W w X x D w x f g x d X x D x f g x d,,,,,,,,

9 Hiding Information The performance of the watermar decoding process is measured by the probability of error, defined as : P e P r w W P z P, r w 9

10 For each value of K, the space y is partitioned into decision regions D 1,..., D M where M W is the no. of possible hidden messages. Decoding errors are due to the uncertainty about the source output X from which the watermared version was obtained. 10

11 Detecting the Watermar For each value of, the watermar detection test can be mathematically defined as a binary hypothesis test in which we have to decide if Z was x generated by the distribution of or the distribution of f x, w,, where X ~ Px and W is modeled as a random variable. 1 x f 0 11

12 Let Z y q z, 1 be the critical region for the watermar detection test performed with, i.e. the set of point in y where s 1 is decided for that ey. The watermar detection test is completely defined by the sets, K 12

13 13 The performance of the watermar detection test is measured by the probabilities of false alarm and detection, defined as : F P P D r r D r r F s Z P P s s P P s Z P P s s P P 1,

14 Suppose there is no distortion during distribution, so Z=Y optimizing the performance of the watermar detection test in terms of PF and PD is in a way equivalent to maximizing the Kullbac- Laibler distance between distributions : P Y s 1, and PY s 0 The maximum achievable distance is limited by the perceptual distortion constraint and entropy of the source. 14

15 The probability of collision between eys and : the probability of deciding s 1 in the watermar detection test for certain ey K 1 when Z has been watermared using a different ey K 2. In the context of copyright protection, this probability should be constrained below a maximum allowed value for all pairs ( K 1, K 2 ) since otherwise the author in possession of K 1 could claim authorship of information watermared by the author who owns. K 2 K 1 K 2 15

16 This constraint imposes a limit to the cardinality of the ey space since the minimum achievable maximum probability of collision between eys increase with the number of eys for fixed P and P. F D 16

17 Attacs In the following discussion we will assume that the attacer has unlimited computation power and that the algorithm for watermaring, detection and decoding are public. The security of the watermaring system relies exclusively on the secret ey K of the copyright owner. 17

18 The Elimination Attac Alternate a watermared source output Y to obtain a negative result s 0 in the watermar detection test for the secret ey used by the legitimate owner. The alteration made by the attacer should not be perceptible, since the resulting output Z will be used as a substitute for the watermared source output Y. 18

19 This constraint can be expressed in mathematical form as an average distortion constraint Ed Z, Y DE or as a maximum distortion constraint dz, Y DE, Z, Y, where d(.,.) is a distortion function and D E is the maximum distortion allowed by the attacer. 19

20 The Elimination Attac can be represented by a game-theoretic model : Given a certain watermared source output Y, the attacer will choose the point Z y, subject to the distortion constraint, which maximizes his probability of success. 20

21 Under a maximum distortion constraint, this maximum probability of success for a given Y can be expressed as P E Y Z : d max Z, Y D E K Y 1 qz, K After averaging out over y, the average probability of success in the elimination attac is max P E P Y P K Y 1 q Z, K Z : d Z, Y D Y E P 21

22 We can model the transformation made by the attacer as a channel with conditioned pdf P Z. Y Then the optimal elimination strategy can be seen as a worst-case channel P Z in the sense that it Y minimizes the P D for given critical regions and watermaring function. f 1 Note that the attacer is limited to those channels which satisfy the average distortion constraint. 22

23 The minimum achievable function of. D E is a non-increasing The optimum watermaring strategy consists in choosing the watermaring function f 1 and the critical regions maximizing the minimum P D achievable by the attacer through the choice of a channel P Z Y.Hence, the design of the watermaring system is a robust hypothesis testing problem. P D 23

24 The Corruption Attac The attacer is not interested in eliminating the watermar, but increasing the probability of error in the watermar decoding process. 24

25 Cryptographic Security The securing level of the system can be measured by the uncertainty about the ey given a watermared source output Y. Using an information-theoretical terminology, this uncertainty is the conditioned entropy HK Y, also called equivocation. 25

26 Size of Key Space A minimum cardinality of the ey space K is a necessary condition for specifying the equivocation HK Y. Increasing the equivocation helps in increasing the robustness against elimination attacs. However, increasing the number of available eys also increases the probability of collision among eys. Therefore, if we specify a maximum allowable probability of collision, this constraint will impose a limit on the maximum number of eys. 26

27 Summary Decoding of hidden information is affected by uncertainty due to the source output (not available at the receiver), distortion and attacs. We can thin that there is a channel between W and Z which can be characterized by a certain capacity. Watermaring and watermar detection under a constrained maximum probability of collision between eys can be seen as an application of identification via channels, with additional constraints derived from the limited admissible perceptual distortion in the watermaring process. The combination of watermar detection and data hiding can be related to the theory of identification plus transmission codes. 27

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