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

Outline Forensic watermarking ◦ collusion attacks q-ary Tardos scheme Density function of "scores" ◦ convolution ◦ series expansion ◦ numerics Open problems 2

Forensic Watermarking EmbedderDetector original content payload content with hidden payload WM secrets payload original content Payload = some secret code indentifying the recipient ATTACK 3

Collusion attacks ABAC CAAA ABAB ACAC ABAB AABCABC "Coalition of pirates" Symbols received by pirates Symbols allowed “Restricted Digit Model” 4

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 5

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 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 Arbitrary alphabet size q Dirichlet distribution F ABCB ACBA BBAC BABA ABAC CAAA ABAB 6

Tardos scheme (cont.) Tracing: Attackers output symbol y i in segment i: Every user gets a score Sum of scores per content segment User is "accused" if score exceeds threshold g 0 (p) p g 1 (p) p 7 For innocent user: E[score]=0 and E[score 2 ]=1

Accusation probabilities m = code length c = #pirates μ = E[coalition score per segment] Pirates want to minimize μ and make the innocent tail longer Curve shapes depend on:  alphabet size q  F, g 0, g 1  Code length  #pirates  Pirate strategy CLT: Big m  curves go to Gaussian Method to compute innocent curve [Simone+Škorić 2010] threshold total score (scaled) innocent guilty 8 S/√m

Finding the innocent score pdf 1.Find pdf of innocent score in one segment. φ (u) 2.Use convolution property of characteristic functions. 9

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

Single-segment pdf 11

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

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! 13

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

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

Power law in the tails 16

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

Combine data for innocent and guilty 18

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 score or multiple scores simultaneously Avoid the series expansion altogether? 19