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!