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Asymptotic fingerprinting capacity for non-binary alphabets

Asymptotic fingerprinting capacity for non-binary alphabets . Dion Boesten, Boris Š kori ć. Outline. Introduction q-ary Tardos scheme Fingerprinting capacity Asymptotic solutions Proof of non-binary case Discussion. Forensic watermarking.

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Asymptotic fingerprinting capacity for non-binary alphabets

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  1. Asymptotic fingerprinting capacity for non-binary alphabets Dion Boesten, Boris Škorić

  2. Outline • Introduction • q-ary Tardos scheme • Fingerprinting capacity • Asymptotic solutions • Proof of non-binary case • Discussion Department of Mathematics & Computer science

  3. Forensic watermarking • Aim: discourage unauthorized distribution of digital content • Watermark consists of two layers: • Coding layer: determines which messages to embed • WM layer: hides the messages in the content • Coding layer history: • Pre Tardos (-2003): highly deterministic • Post-Tardos (2003-): fully probabilistic, optimal asymptotic code length Department of Mathematics & Computer science

  4. Forensic watermarking originalcontent originalcontent watermarked content unique watermark unique watermark Detector Embedder Attack

  5. q-ary Tardos scheme content segments • Code generation • Biases drawn from distribution F • Code entries generated per segment using bias • Coalition attack • Coalition size • Attack is limited by Restricted Digit Model • Special case is Marking Assumption symbol biases n users pirates allowed attack symbols Department of Mathematics & Computer science

  6. Accusation • Aim: Detect at least 1 of the pirates • Accusation procedure • User code words are compared with pirated watermark • Each user receives a score • If exceeds a threshold then user is considered guilty • Error probabilities • False positive: innocent user is accused • False negative: none of the pirates are accused Department of Mathematics & Computer science

  7. Collusion channel pirate code words allowed attack symbols Attack strategy Attack strategy • Optimal attack is segment independent • Count frequency of occurred symbols • Choose output symbol probabilistically: • Example: Interleaving attack • Attack can be seen as noise on a communication channel Department of Mathematics & Computer science

  8. Fingerprinting capacity - + Department of Mathematics & Computer science • Mutual Information • We know • We want to know (equivalent with pirates’ identity) • Fingerprinting game • Payoff function is • Content owner chooses bias distribution • Pirates decide on a strategy • Fingerprinting capacity is derived as:

  9. Importance of capacity code length # of users • Capacity provides a lower bound on required code lengths • Rate of the code is: • A reliable code should have : / name of department

  10. Asymptotic solutions • Asymptotic limit # of pirates • Binary alphabet () • Solution found by Huang and Moulin (2010) • (Arcsine distribution) • (Interleaving attack) • Non-binary alphabet () • We solved non-binary case Department of Mathematics & Computer science

  11. Proof of non-binary case (1/4) As we assume: • The random variable becomes continuous in with expected value • The attack strategy can be approximated by continuous functions : Department of Mathematics & Computer science

  12. Proof of non-binary case (2/4) • We have • Taylor expansion of strategy: • Expand payoff function: Department of Mathematics & Computer science

  13. Proof of non-binary case (3/4) • Reversal of max-min game • By Sion’s minimax theorem: • Max-min is equal to min-max only by optimal value Department of Mathematics & Computer science

  14. Proof of non-binary case (4/4) • Solving has two parts: • We prove for any attack strategy : • The Interleaving attack has: Department of Mathematics & Computer science

  15. More details of the proof • How to prove ? • with the Jacobian matrix of the mapping • Both p and g are probability vectors so / name of department

  16. More details of the proof An infinitesimal surface element is related to the corresponding element by a factor of The total surface area is equal or larger to / name of department

  17. More details of the proof • If there must be a point where • Theorem (AM-GM inequality): • If then / name of department

  18. Discussion • is an increasing function of • Advantageous to use larger • Actual implementation and attack options determine achievable • Future work: • Solve Max-min game to obtain optimal asymptotic strategies • Find capacity for different attack models Department of Mathematics & Computer science

  19. Questions? Department of Mathematics & Computer science

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