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Asymptotic fingerprinting capacity for non-binary alphabets

This study explores the asymptotic fingerprinting capacity for non-binary alphabets, focusing on the q-ary Tardos scheme. It addresses the necessity for effective forensic watermarking to deter unauthorized content distribution. We delve into coding and watermark layers, user collaboration in attacks, and optimal strategies against them. The paper constructs a mathematical foundation, proving results for non-binary cases, while discussing implications for content owners and pirates. Our findings underscore the importance of capacity in determining code lengths and developing future research directions.

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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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