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

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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

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

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

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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

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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

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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

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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

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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

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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

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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

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

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Results Gaussian approximation 13

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Results (cont.) Gaussian approximation 14

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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

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