Statistical properties of Tardos codes

1. Statistical properties of Tardos codes Boris Škorić and Antonino Simone Eindhoven University of Technology Stochastics Seminar, 28 Nov. 2012

2. Outline • Forensic watermarking • collusion attacks • q-aryTardos scheme • Density function of "scores" • convolution • series expansion • numerics • Open problems

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. Collusion attacks "Coalition of pirates" Symbols received by pirates “Restricted Digit Model” Symbols allowed

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) ⇒ longer code AND Limited bandwidth available for watermark

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

7. 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 p g0(p) g1(p) For innocent user:E[score]=0 and E[score2]=1 p

8. Accusation probabilities Pirates want to minimize μand make the innocent tail longer m = code length c = #pirates μ = E[coalition score per segment] threshold • Curve shapes depend on: • alphabet size q • F, g0, g1 • Code length • #pirates • Pirate strategy guilty innocent S/√m total score (scaled) CLT: Big m  curves go to Gaussian Method to compute innocent curve [Simone+Škorić 2010]

9. Finding the innocent score pdf Find pdf of innocent score in one segment.φ(u) Use convolution property of characteristic functions.

10. Innocent score pdf (2) Finding the single-segment pdf: attack strategy

11. Single-segment pdf

12. Innocent score pdf (3) The Fourier transform: hypergeometric

13. 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!

14. Innocent score pdf (5) • Less direct approach for finding False Positive prob: • Still use same starting point • ... but do Edgeworth-like expansion Hermite function Gaussian tail • ... and then pray for numerical stability

15. Numerical results on False Positive probs. Convergence not enough terms

16. Power law in the tails

17. Score pdf for one guilty user • Same approach, minor differences: • Nonzero mean (strategy dependent) • Variance depends on attack strategy

18. Combine data for innocent and guilty

19. 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?