Asymptotic channel capacity of collusion resistant watermarking for no ir. Dion Boesten, dr. Boris Škorić Introduction Forensic 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 • Accusation An accusation algorithm compares the detected watermark with the matrix to produce candidate Forensic watermarking watermark embedding watermark detector original content unique watermark watermarked content unique watermark original content 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 1,2,3 watermark after attack n users code 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 attack strategy 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: C 2 1 2 2 ln2 − 1 2 2 ln 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. Discussion The fingerprinting capacity is an increasing function of . Hence it makes sense to switch to higher alphabets if the embedding scheme allows this.
Asymptotic channel capacity of collusion resistant watermarking for non-binary alphabets
Feb 23, 2016
original content. original content. w atermarked content. u nique watermark. unique watermark. w atermark d etector. w atermark embedding. Asymptotic channel capacity of collusion resistant watermarking for non-binary alphabets ir. Dion Boesten, dr. Boris Škorić. Introduction - PowerPoint PPT Presentation
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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.
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