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Asymptotically false- positive-maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011, May 2011
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Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

Mar 31, 2015

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Page 1: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

Asymptotically false-positive-maximizing attack on non-binary Tardos codes

Antonino Simone and Boris Škorić

Eindhoven University of Technology

IH 2011, May 2011

Page 2: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

2

OutlineForensic watermarking

◦Collusion attacksq-ary Tardos schemeNew parameterization of attack

strategyAccusation-minimizing attackPerformance of the Tardos scheme

◦False accusation probabilityResults & Summary

Page 3: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

3

Forensic Watermarking

Embedder Detector

originalcontent

payload

content withhidden payload

WM secrets

WM secrets

payload

originalcontent

Payload = some secret code indentifying the recipient

ATTACK

Page 4: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

4

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”

Page 5: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

5

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

Page 6: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

6

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

• Symbol-symmetric

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: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

7

Tardos scheme (cont.)Accusation:

• Every user gets a score

• User is accused if score > threshold

• Sum of scores per content segment

• Given that pirates create y in segment i:

• Symbol-symmetric

g0(p)

g1(p)

p

p

Page 8: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

8

Accusation probabilitiesm = code length

c = #pirates

μ̃E = expected coalition score per segment

Pirates want to minimize μ̃E and make the innocent tail longer

Curve shapes depend on: F, g0, g1 (fixed ‘a

priori’) Code length # pirates Pirate strategy

Method to compute innocent curve [Simone+Škorić 2010]Big m innocent curve goes to Gaussian

threshold

total score (scaled)

innocent guilty

Page 9: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

9

New parameterization of attack strategy

Symbol-symmetric only symbol occurrences matter

Notation: α = # α in segmentc pirates α α = c

For every segment:

New attack parameterization that does not refer to symbols:

Page 10: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

10

New parameterization of attack strategy (cont.)

Due to the marking assumption, K0=0 and Kc=1

Kb can be pre-computed faster computation

Thanks to the new parameterization, we can write

Which strategy minimizes μ̃E?

])1[(

)2/1]1[(

)(

)2/1()1(

2

1)(

)(]Pr[~1

qbc

qbc

b

bcq

c

bbT

bTKbqc

bb

Page 11: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

11

μ̃E-minimizing attack

For each , the attack outputs the symbol y s. t. its occurrence value y minimizes T(b) (i. e. T(y)T() for each )

Page 12: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

12

T(b) analysisStrong influence of parameter

More interesting case:

Majority voting

Minority voting

Page 13: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

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ResultsGaussian approximation

Page 14: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

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Results (cont.)Gaussian approximation

Page 15: Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011,

15

SummaryResults: simple decoder accusation method in the

Restricted Digit Model new parameterization of the attack strategy μ̃E-minimizing attack is the strongest attack in

asymptotic regime◦ not optimal attack for small coalitions

parameter has a strong effect For q>2 code length becomes better than for q=2,

but only if c is large enough! The larger q is, the larger c must be to obtain a

code shorter than the case q=2

Thank you for your attention!