Accusation probabilities in Tardos codes

1. Accusation probabilities in Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology WISSec 2010, Nov 2010

2. Outline • Introduction to forensic watermarking • Collusion attacks • Aim • Tardos scheme • q-ary version • Properties • Performance of the Tardos scheme • False accusation probability • Results & Summary

3. originalcontent originalcontent content withhidden payload payload WM secrets payload WM secrets Detector Embedder Forensic Watermarking ATTACK Payload = some secret code indentifying the recipient

4. "Coalition of pirates"  = "detectable positions" pirate #1 1 1 1 0 1 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 #2 1 0 1 0 1 0 1 0 0 0 1 1 #3 1 1 1 0 0 0 1 1 0 0 0 1 #4 AttackedContent 1 0/1 1 0 0/1 0 1 0/1 0/1 0 0/1 1 Collusion attacks

5. Aim Trace at least one pirate from detected watermark BUT Resist large coalition  longer code Low probability of innocent accusation (FP) (critical!)  longer code Low probability of missing all pirates (FN) (not critical)  longer code AND Limited bandwidth available for watermarking code

6. q-aryTardos scheme (2008) m content segments biases Symbol biases drawn from distribution F embedded symbols • Arbitrary alphabet size q • Dirichletdistribution F n users c pirates Symbols allowed =y watermark after attack

7. Tardos scheme continued • Accusation: • 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

8. Properties of the Tardos scheme • Asymptotically optimal • m  c2 for large coalitions, for every q • Previously best m  c4 • Proven: power ≥ 2 • Random code book • No framing • No risk to accuse innocent users if coalition is larger than anticipated • F, g0 and g1 chosen ‘ad hoc’ (can still be improved)

9. Accusation probabilities Result: majority voting minimizes u m = code length c = #pirates u = avg guilty score Pirates want to minimize u and make longer the innocent tail threshold • Curve shapes depend on: • F, g0, g1 (fixed ‘a priori’) • Code length • # pirates • Pirate strategy guilty innocent u total score (scaled) Central Limit Theorem  asymptotically Gaussian shape (how fast?) 2003  2010: innocent accusation curve shape unknown… till now!

10. Approach Fourier transform property: • Steps: • S = iSi • Si  •  = pdf of total score S • S   = InverseFourier[ ] • Compute • Depends on strategy • New parameterization for attack strategy • Compute • Taylor • Taylor • Taylor

11. Main result: false accusation probability curve threshold/√m Example: exact FP majority voting attack Result from Gaussian log10FP FP is 70 times less than Gaussian approx in this example But  Code 2-5% shorter than predicted by Gaussian approx

12. Summary Results: • 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!