Asymptotic channel capacity of collusion resistant watermarking for non-binary alphabets

Asymptotic channel capacity of collusion resistant watermarking for non-binary alphabetsir. Dion Boesten, dr. Boris Škorić

IntroductionForensic watermarking is a technique to combat the unauthorized distribution of digital content. Each copy is embedded with an imperceptible and unique watermark. The main challenge in this area is to create codes that are resistant to collusion attacks. In a collusion attack a group of users (pirates) compare their copies and try to create an untraceable copy. Information theory can provide a lower bound on the required code length of a reliable collusion resistant code.

Bias based codes• Code generation

Bias vectors are drawn from a distribution Codeword symbol is generated stochasticallly using bias :

• Attack Pirates choose one of the allowed symbols in each segment By the Marking Assumption they cannot modify a segment in which

they all received the same symbol• AccusationAn accusation algorithm compares the detected watermark with the matrix to produce candidate coalitions. This is most often done by assigning scores to individual users and then applying a threshold procedure (Tardos).

Forensic watermarking

watermarkembedding

watermarkdetector

originalcontent

uniquewatermark

watermarkedcontent unique

watermark

originalcontent

pirate attack

1 2 3 2

1 3 2 1

2 3 1 3

2 1 2 1

1 2 1 3

3 1 1 1

2 2 1 2

1,2,3 1,2 1 1,2,3watermark after attack

n userscode matrix X

m content segments

bias vectors biases

c pirates

allowed attack symbols

Attack channel

• Optimal attack is segment independent and can be seen as a noisy communication channel

• counts the number of symbols that the pirates received• Channel output is generated stochastically:

counter variable

output symbol

pirate attackstrategy

Channel capacity• The mutual information measures how much

information reveals about the identity of the pirates (equivalent with )

• The channel capacity is derived as the optimal value of a max-min game:

• Asymptotic solution () Binary case was solved by Huang and

Moulin[2010]:

Non-binary case was solved by us:

C21

2𝑐2ln 2

𝐶𝑞𝑞−1

2𝑐2 ln𝑞

Proof outline• Sion’s Theorem: • Continuum approximation of the strategy: • Taylor of around and expanding the payoff function yields a

result containing the Fisher Information of conditioned on . This result can also be expressed as where is the Jacobian matrix .

• We use the inequality and topological properties of the mapping to bound

• Finally we identify a which achieves this lower bound.

DiscussionThe fingerprinting capacity is an increasing function of . Hence it makes sense to switch to higher alphabets if the embedding scheme allows this.