1. Information Theoretical Analysis of Digital Watermarking
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
: a rv. which indicates whether X will be watermarked.
The variable S is introduced in the model only to provide the possibility of expressing mathematically the existence or non-existence of a watermark in a simple way.

3. K : a secret key defined on a discrete alphabet k.
S=1 : (watermarked version)
: (The output of the watermarking function )
S=0 : (non-watermarked version) :
The output of the watermarking function depends on the value of K, a secret key which uniquely identifies the copyright owner.

4. General model of a watermarking system
X k
W Y Z K
General model of a watermarking system g fs ψ q

5. The watermarked version Y then passes through a noisy channel and is transformed into This channel models both unintentional distortions suffered by Y and attacks aimed at deleting or corrupting the watermark information.
In both cases we assume that the secret key is not known, so the noisy channel can be defined by the distribution which is independent of K.

6. Finally, Z is processed to obtain a point which will be used by the recipient instead of X.
There are two tests that can serve to verify the ownership of Z :
the watermark detection test
the watermark decoding test
the detection test is used to obtain an estimate of S (to decide whether Z has been watermarked using k)
the decoding test is used to obtain an estimate of W.

7. Imperceptibility :
Let be a perceptually significant distortion. A watermarking system must guarantee that the functions , and g introduce imperceptible alternations with respect to X.
With expectations taken wrt. X, W, K,
(Mean Distortion Consraints)

8. or
(Maximum constraints)

9. Hiding Information
The performance of the watermark decoding process is measured by the probability of error, defined as :

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

11. Detecting the Watermark
For each value of k, the watermark detection test can be mathematically defined as a binary hypothesis test in which we have to decide if Z was generated by the distribution of or the distribution of , where and W is modeled as a random variable.

12. Let be the critical region for the watermark detection test performed with k, i.e. the set of point in y where is decided for that key.
The watermark detection test is completely defined by the sets

13. The performance of the watermark detection test is measured by the probabilities of false alarm and detection , defined as :

14. Suppose there is no distortion during distribution, so Z=Y optimizing the performance of the watermark detection test in terms of and is in a way equivalent to maximizing the Kullback-Laibler distance between distributions :
and
The maximum achievable distance is limited by the perceptual distortion constraint and entropy of the source.

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

16. This constraint imposes a limit to the cardinality of the key space since the minimum achievable maximum probability of collision between keys increase with the number of keys for fixed and

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

18. The Elimination Attack
Alternate a watermarked source output Y to obtain a negative result in the watermark detection test for the secret key used by the legitimate owner.
The alteration made by the attacker should not be perceptible, since the resulting output Z will be used as a substitute for the watermarked source output Y.

19. This constraint can be expressed in mathematical form as an average distortion constraint or as a maximum distortion constraint where d(.,.) is a distortion function and is the maximum distortion allowed by the attacker.

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

21. Under a maximum distortion constraint, this maximum probability of success for a given Y can be expressed as
After averaging out over y, the average probability of success in the elimination attack is

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

23. The minimum achievable is a non-increasing function of .
The optimum watermarking strategy consists in choosing the watermarking function and the critical regions maximizing the minimum achievable by the attacker through the choice of a channel Hence, the design of the watermarking system is a robust hypothesis testing problem.

24. The Corruption Attack
The attacker is not interested in eliminating the watermark, but increasing the probability of error in the watermark decoding process.

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

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

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