Asymptotic fingerprinting capacity in the Combined Digit Model Dion Boesten and Boris Škorić presented by Jan-Jaap Oosterwijk

Outline forensic watermarking collusion attack models: Restricted Digit Model and Combined Digit Model bias-based codes fingerprinting capacity large coalition asymptotics Previous results: Restricted Digit Model New contribution: Combined Digit Model

Forensic watermarking EmbedderDetector original content unique watermark watermarked content unique watermark original content Attack

Collusion attacks ABCB ACBA BBAC BABA ABAC CAAA ABAB n users ABAC CAAA ABAB Simplifying assumption: segments into which q-ary symbols can be embedded collusion attack: c attackers pool their resources m content segments

Attack models: Restricted Digit Model (RDM) "Marking assumption": can't produce unseen symbol Restricted Digit Model: choose from available symbols ABCB ACBA BBAC BABA ABAC CAAA ABAB ABAC CAAA ABAB m content segments allowed symbols ACAC ABAB AABCABC c attackers

Attack models: Combined Digit Model (CDM) [BŠ et al. 2009] More realistic Allows for signal processing attacks mixing noise alphabet Q received Ω ⊆ Q mixed: Ψ ⊆ Ω detected: W attack symbol detection probability: r 1-r 1-t |ψ| t Noise parameter r. Mixing parameters t 1 ≥ t 2 ≥ t 3...

Bias-based codes [Tardos 2003] ABCB ACBA BBAC BABA ABAC CAAA ABAB symbol biases ABAC CAAA ABAB Code generation Biases drawn from distribution F Code entries generated per segment j using the bias: Pr[X ij = α] = p jα. Attack Coalition size c. Same strategy in each segment In Combined Digit Model: strategy = choice of subset Ψ ⊆ Ω, possibly nondeterministic. Accusation algorithm for finding at least one attacker, based on distributed and observed symbols. Ω={A,B} Allowed Ψ: {A}, {B}, {A,B}

Collusion attack viewed as malicious noise Noisy communication channel From symbol embedding to detection Coalition attack causes "noise" Channel capacity Apply information theory Rate of a tracing code: R = (log q n)/m Capacity C = max. achievable rate. Fundamental upper bound. Results for Restricted Digit Model, and #attackers → ∞ Huang&Moulin 2010 Binary codes (q=2): Boesten&Škorić 2011 Arbitrary alphabet size: n = #users m = #segments q = alphabet size

Capacity for the Combined Digit Model The math Look at one segment Define counters Σ α = #attackers who receive α Parametrization of the attack strategy: Capacity: p = bias vector F = prob. density for p W = set of detected symbols H(Σ) H(W) I(W;Σ) +- F θ

CDM capacity: further steps Apply Sion's theorem "Value" of max-min and min-max game is the same! Limit c → ∞: Σ very close to cp Taylor expansion in Σ/c – p Re-paramerization γ: mapping from q-dim. hypersphere to (2 q -1)-dim. hypersphere. Jacobian J Pay-off function Tr(J T J)

CDM capacity: constraints Looks like beautiful math, but... nasty constraint on the mapping γ We did not dare to try q>2 Binary case: Constrained geodesics

CDM capacity: numerical results for q=2 Part of the graphs we understand intuitively Stronger attack options => lower capacity Near (r=0, t 1 =1) RDM-like behaviour; weak dependence on t 2 Away from RDM we have little intuition

Summary Asymptotic capacity for the Combined Digit Model Partly the same exercise as in Restricted Digit Model Find optimal hypersphere → hypersphere mapping But... higher-dimensional space nasty constraint on the mapping Numerics for binary alphabet constrained geodesics in 2 dimensions graphs show how attack parameters (r, t 1, t 2 ) affect capacity useful for code design Future work (perhaps...) change attack model to get analytic results

Questions?