Asymptotically false-positive- maximizing attack on non-binary Tardos codes Antonino Simone and Boris Škorić Eindhoven University of Technology IH 2011, May 2011

Outline Forensic watermarking ◦ Collusion attacks q-ary Tardos scheme New parameterization of attack strategy Accusation-minimizing attack Performance of the Tardos scheme ◦ False accusation probability Results & Summary 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) (not critical)  longer code AND Limited bandwidth available for watermarking code 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 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 Symbol-symmetric ABCB ACBA BBAC BABA ABAC CAAA ABAB 6

Tardos scheme (cont.) Accusation: Every user gets a score User is accused if score > threshold Sum of scores per content segment Given that pirates create y in segment i: Symbol-symmetric g 0 (p) g 1 (p) p p 7

Accusation probabilities m = code length c = #pirates μ ̃ = expected coalition score per segment Pirates want to minimize μ ̃ and make the innocent tail longer Curve shapes depend on:  F, g 0, g 1 (fixed ‘a priori’)  Code length  # pirates  Pirate strategy Method to compute innocent curve [Simone+Škorić 2010] Big m  innocent curve goes to Gaussian threshold total score (scaled) innocent guilty 8

New parameterization of attack strategy Symbol-symmetric  only symbol occurrences matter Notation:  α = # α in segment c pirates   α  α = c For every segment: New attack parameterization that does not refer to symbols: 9

New parameterization of attack strategy (cont.) Due to the marking assumption, K 0 =0 and K c =1 K b can be pre-computed  faster computation Thanks to the new parameterization, we can write Which strategy minimizes μ ̃ ? 10

μ ̃ -minimizing attack For each , the attack outputs the symbol y s. t. its occurrence value  y minimizes T(b) (i. e. T(  y )  T(   ) for each  ) 11

T(b) analysis Strong influence of  parameter More interesting case: Majority voting Minority voting 12

Results Gaussian approximation  13

Results (cont.) Gaussian approximation  14

Summary Results: simple decoder accusation method in the Restricted Digit Model new parameterization of the attack strategy μ ̃ -minimizing attack is the strongest attack in asymptotic regime ◦ not optimal attack for small coalitions  parameter has a strong effect For q>2 code length becomes better than for q=2, but only if c is large enough! The larger q is, the larger c must be to obtain a code shorter than the case q=2 Thank you for your attention! 15