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Asymptotic fingerprinting capacity for non-binary alphabets
Dion Boesten, Boris Škorić
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Department of Mathematics & Computer science 22-04-2023 PAGE 2
Outline
• Introduction• q-ary Tardos scheme• Fingerprinting capacity
• Asymptotic solutions• Proof of non-binary case• Discussion
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Department of Mathematics & Computer science 22-04-2023
Forensic watermarking
• Aim: discourage unauthorized distribution of digital content
• Watermark consists of two layers:• Coding layer: determines which messages to embed• WM layer: hides the messages in the content
• Coding layer history:• Pre Tardos (-2003): highly deterministic• Post-Tardos (2003-): fully probabilistic, optimal asymptotic
code length
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Forensic watermarking
Embedder Detector
originalcontent
unique watermark
watermarkedcontent unique
watermark
originalcontent
Attack
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Department of Mathematics & Computer science 22-04-2023
q-ary Tardos scheme
A B C B
A C B A
B B A C
B A B A
A B A C
C A A A
A B A B
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symbol biases
content segments
n users
Code generation• Biases drawn from
distribution F• Code entries generated
per segment using bias
Coalition attack• Coalition size • Attack is limited by
Restricted Digit Model• Special case is Marking
Assumption pirates
A B A CC A A AA B A B
allowedattack
symbols
AC
AB
A ABC
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Department of Mathematics & Computer science 22-04-2023
Accusation
• Aim: Detect at least 1 of the pirates
• Accusation procedure• User code words are compared with pirated watermark• Each user receives a score • If exceeds a threshold then user is considered guilty
• Error probabilities• False positive: innocent user is accused• False negative: none of the pirates are accused
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Department of Mathematics & Computer science 22-04-2023
Collusion channel
Attack strategy• Optimal attack is segment
independent• Count frequency of occurred
symbols • Choose output symbol
probabilistically:
• Example: Interleaving attack • Attack can be seen as noise
on a communication channel
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A B A CC A A AA B A B
AC
AB
A ABC
piratecodewords
allowedattacksymbols
𝚺=(120)Attack
strategy𝚺 𝑌
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Department of Mathematics & Computer science 22-04-2023
Fingerprinting capacity
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• Mutual Information• We know • We want to know (equivalent
with pirates’ identity)
• Fingerprinting game• Payoff function is • Content owner chooses bias
distribution • Pirates decide on a strategy • Fingerprinting capacity is
derived as:
𝐹𝜽
𝐻 (𝚺)𝐻 (𝑌 ) 𝐼 (𝑌 ;𝚺)
𝑰+-
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/ name of department 22-04-2023
Importance of capacity
• Capacity provides a lower bound on required code lengths
• Rate of the code is:
• A reliable code should have :
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code length # of users
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Department of Mathematics & Computer science 22-04-2023
Asymptotic solutions
• Asymptotic limit # of pirates
• Binary alphabet ()• Solution found by Huang and Moulin (2010)
− (Arcsine distribution)− (Interleaving attack)
• Non-binary alphabet ()• We solved non-binary case
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Department of Mathematics & Computer science 22-04-2023
Proof of non-binary case (1/4)
As we assume:• The random variable becomes continuous in
with expected value • The attack strategy can be approximated by
continuous functions :
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Department of Mathematics & Computer science 22-04-2023
Proof of non-binary case (2/4)
• We have • Taylor expansion of strategy:
• Expand payoff function:
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Department of Mathematics & Computer science 22-04-2023
Proof of non-binary case (3/4)
• Reversal of max-min game• By Sion’s minimax theorem:
• Max-min is equal to min-max only by optimal
value
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Department of Mathematics & Computer science 22-04-2023
Proof of non-binary case (4/4)
• Solving has two parts:
• We prove for any attack strategy :
• The Interleaving attack has:
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min𝒈max𝒑
𝑇 (𝒑 )=𝑞−1
𝐶𝑞=𝑞−12𝑐2 ln𝑞
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More details of the proof
• How to prove ?• with the Jacobian matrix of the mapping
• Both p and g are probability vectors so
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/ name of department 22-04-2023
More details of the proof
• An infinitesimal surface element is related to the corresponding element by a factor of
• The total surface area is equal or larger to
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/ name of department 22-04-2023
More details of the proof
• If there must be a point where
• Theorem (AM-GM inequality): • If then
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Department of Mathematics & Computer science 22-04-2023
Discussion
• is an increasing function of • Advantageous to use larger • Actual implementation and attack options
determine achievable • Future work:• Solve Max-min game to obtain optimal
asymptotic strategies• Find capacity for different attack models
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Department of Mathematics & Computer science 22-04-2023
Questions?
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