Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of...
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![Page 1: Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of Technology Stochastics Seminar, 28 Nov. 2012.](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d055503460f949d8f62/html5/thumbnails/1.jpg)
Statistical properties of Tardos codes
Boris Škorić and Antonino Simone
Eindhoven University of Technology
Stochastics Seminar, 28 Nov. 2012
![Page 2: Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of Technology Stochastics Seminar, 28 Nov. 2012.](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d055503460f949d8f62/html5/thumbnails/2.jpg)
OutlineForensic watermarking
◦ collusion attacksq-ary Tardos schemeDensity function of "scores"
◦ convolution◦ series expansion◦ numerics
Open problems
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Forensic Watermarking
Embedder Detector
originalcontent
payload
content withhidden payload
WM secrets
WM secrets
payload
originalcontent
Payload = some secret code indentifying the recipient
ATTACK
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Collusion attacks
A B A C
C A A A
A B A B
AC
AB
A ABC
"Coalition of pirates"Symbols received by pirates
Symbols allowed
“Restricted Digit Model”
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Aim
Trace at least one pirate from detected watermark
BUTResist large coalition
⇒ longer codeLow probability of innocent accusation (FP) (critical)
⇒ longer codeLow probability of missing all pirates (FN) ⇒ longer code ANDLimited bandwidth available for watermark
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n users
embeddedsymbols
m content segments
Symbols allowed
Symbol biases
drawn from distribution
F
watermarkafter attack
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
AC
AB
A ABC
p1A
p1B
p1C
p2A
p2B
p2C
piA
piB
piC
pm
A
pm
B
pm
C
c pirates
q-ary Tardos scheme
• Arbitrary alphabet size q
• Dirichlet distribution F
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
![Page 7: Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of Technology Stochastics Seminar, 28 Nov. 2012.](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d055503460f949d8f62/html5/thumbnails/7.jpg)
Tardos scheme (cont.)Tracing:
• Attackers output symbol yi in segment i:
• Every user gets a score
• Sum of scores per content segment
• User is "accused" if score exceeds threshold
g0(p)
p
g1(p)
p
For innocent user:E[score]=0 and E[score2]=1
![Page 8: Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of Technology Stochastics Seminar, 28 Nov. 2012.](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d055503460f949d8f62/html5/thumbnails/8.jpg)
Accusation probabilities
m = code length
c = #pirates
μ = E[coalition score per segment]
Pirates want to minimize μand make the innocent tail longerCurve shapes depend
on: alphabet size q F, g0, g1
Code length #pirates Pirate strategy
CLT: Big m curves go to GaussianMethod to compute innocent curve [Simone+Škorić 2010]
threshold
total score (scaled)
innocent guilty
S/√m
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Finding the innocent score pdf
1. Find pdf of innocent score in one segment.φ(u)
2. Use convolution property of characteristic functions.
€
˜ ϕ (k) = [Fϕ ](k)
€
˜ ρ S (k) = [Fρ S ](k) = [ ˜ ϕ (k)]m
€
ρS / m
€
ρS / m
= mρ S
€
˜ ρ S / m
(k) = ˜ ρ S (k / m ) = [ ˜ ϕ ( k
m)]m
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Innocent score pdf (2)
Finding the single-segment pdf:
attack strategy
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Single-segment pdf
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Innocent score pdf (3)
The Fourier transform:
hypergeometric
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Innocent score pdf (4)
Direct approach for finding False Positive prob:
Prob[S>Z] =
Z/√m
Try numerical computation of the k-integral.
Problem: numerical instability!
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Innocent score pdf (5)
Less direct approach for finding False Positive prob:• Still use same starting point
• ... but do Edgeworth-like expansion
Gaussian tail Hermite function
• ... and then pray for numerical stability14
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Numerical results on False Positive probs.Convergence
not enough terms
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Power law in the tails
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Score pdf for one guilty user
Same approach, minor differences:• Nonzero mean (strategy dependent) • Variance depends on attack strategy
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Combine data for innocent and guilty
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Open questions / future work
• Better understanding of the convergence
• Reduce the reliance on "prayer"
• Strategy-independent bounds
• avoid re-doing everything for each strategy
• Do the whole exercise for the coalition scoreor multiple scores simultaneously
• Avoid the series expansion altogether?
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