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

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

Forensic Watermarking EmbedderDetector original content payload content with hidden payload WM secrets payload original content Payload = some secret code indentifying the recipient ATTACK

Collusion attacks "Coalition of pirates" 1 pirate #1 Attacked Content / #2 #3 #4 = "detectable positions"

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

n users embedded symbols m content segments Symbols allowed Symbol biases drawn from distribution F watermark after attack ABCB ACBA BBAC BABA ABAC CAAA ABAB biases ACAC ABAB AABCABC p 1A p 1B p 1C p 2A p 2B p 2C p iA p iB p iC p mA p mB p mC c pirates q-ary Tardos scheme (2008) Arbitrary alphabet size q Dirichlet distribution F =y ABCB ACBA BBAC BABA ABAC CAAA ABAB

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

Properties of the Tardos scheme Asymptotically optimal ◦ m  c 2 for large coalitions, for every q ◦ Previously best m  c 4 ◦ Proven: power ≥ 2 Random code book No framing ◦ No risk to accuse innocent users if coalition is larger than anticipated F, g 0 and g 1 chosen ‘ad hoc’ (can still be improved)

Accusation probabilities m = code length c = #pirates u = avg guilty score Pirates want to minimize u and make longer the innocent tail Curve shapes depend on:  F, g 0, g 1 (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

Approach Fourier transform property: Steps: 1.S =  i S i Si Si   = pdf of total score S S   = InverseFourier[ ] 2. 3.Compute Depends on strategy New parameterization for attack strategy 4.Compute 5. 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 log 10 FP

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!