Asymptotic fingerprinting capacity for non-binary alphabets

Asymptotic fingerprinting capacity for non-binary alphabets

Dion Boesten, Boris Škorić

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

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


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


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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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


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+-

/ 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


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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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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• We have • Taylor expansion of strategy:

• Expand payoff function:

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• 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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• 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


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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• 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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• If there must be a point where

• Theorem (AM-GM inequality): • If then

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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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Questions?

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Asymptotic fingerprinting capacity for non-binary alphabets