Accusation probabilities in Tardos codes Antonino Simone and Boris Škorić Eindhoven University of...
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Transcript of Accusation probabilities in Tardos codes Antonino Simone and Boris Škorić Eindhoven University of...
Accusation probabilities in Tardos codes
Antonino Simone and Boris Škorić
Eindhoven University of Technology
WISSec 2010, Nov 2010
OutlineIntroduction to forensic
watermarking◦Collusion attacks◦Aim
Tardos scheme◦q-ary version◦Properties
Performance of the Tardos scheme◦False accusation probability
Results & Summary
Forensic Watermarking
Embedder Detector
originalcontent
payload
content withhidden payload
WM secrets
WM secrets
payload
originalcontent
Payload = some secret code indentifying the recipient
ATTACK
Collusion attacks"Coalition of pirates"
1pirate #1
AttackedContent
1
1
0
0
0
0
1
1
1
10
0
0
0
0
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
1
0
1
0
0
0
0
0
0
1
1
1
1
0
1
1
0
1 0/1 1 0 0/1 0 1 0/1 0/1 0 0/1 1
#2
#3
#4
= "detectable positions"
AimTrace 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) (not critical) longer codeANDLimited bandwidth available for watermarking code
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
biases
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 (2008)
• Arbitrary alphabet size q
• Dirichlet distribution F
=y
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
Tardos scheme continuedAccusation:
• Every user gets a score
• User is accused if score > threshold
• Sum of scores per content segment
• Given that pirates have y in segment i:
• Symbol-symmetric
Properties of the Tardos schemeAsymptotically optimal
◦m c2 for large coalitions, for every q◦Previously best m c4
◦Proven: power ≥ 2Random code bookNo framing
◦No risk to accuse innocent users if coalition is larger than anticipated
F, g0 and g1 chosen ‘ad hoc’ (can still be improved)
Accusation probabilitiesm = code length
c = #pirates
u = avg guilty score
Pirates want to minimize u and make longer the innocent tail
Curve shapes depend on: F, g0, g1 (fixed ‘a
priori’) Code length # pirates Pirate strategy
Central Limit Theorem asymptotically Gaussian shape (how fast?)2003 2010: innocent accusation curve shape unknown… till now!
threshold
total score (scaled)
u
Result: majority voting minimizes u
innocent guilty
ApproachFourier transform property:
Steps:1. S = i Si
Si = pdf of total score SS = InverseFourier[ ]
2.
3. Compute • Depends on strategy• New parameterization for attack strategy
4. Compute5.
• Taylor • Taylor• Taylor
Main result: false accusation probability curve
Example:
majority voting attack
threshold/√m
exact FP
Result from Gaussian
FP is 70 times less than Gaussian approx in this example
But
Code 2-5% shorter than predicted by Gaussian approx
log10FP
SummaryResults: introduced a new parameterization of the attack
strategy majority voting minimizes u first to compute the innocent score pdf
◦ quantified how close FP probability is to Gaussian◦ sometimes better then Gaussian!◦ safe to use Gaussian approx
Future work: study more general attacks different parameter choices
Thank you for your attention!