asymptotic fingerprinting capacity in the combined digit model
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Asymptotic fingerprinting capacity in the Combined Digit Model
Dion Boesten and Boris Škorić
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
Embedder Detector
originalcontent
unique watermark
watermarkedcontent unique
watermark
originalcontent
Attack
Collusion attacks
A B C B
A C B A
B B A C
B A B A
A B A C
C A A A
A B A B
n users
A B A C
C A A A
A B A B
• 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
A B C B
A C B A
B B A C
B A B A
A B A C
C A A A
A B A B
A B A C
C A A A
A B A B
m content segments
allowedsymbols
AC
AB
A ABC
c attackers
Attack models: Combined Digit Model (CDM)
[BŠ et al. 2009]• More realistic • Allows for signal processing attacks
• mixing• noise
alphabetQ
receivedΩ Q⊆
mixed:Ψ Ω⊆
detected:W
attack
symbol detectionprobability:
r
1-r
1-t|ψ|
t |ψ|
Noise parameter r. Mixing parameters t1 ≥ t2 ≥ t3 ...
Bias-based codes [Tardos 2003]
A B C B
A C B A
B B A C
B A B A
A B A C
C A A A
A B A B
symbol biases
content segments
A B A C
C A A A
A B A B
Code generation• Biases drawn
from distribution F• Code entries generated
per segment j using the bias:
Pr[Xij = α] = pjα.
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 = (logq 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 2011Arbitrary alphabet size:
€
C2 =1
2c 2 ln2
€
Cq =q−1
2c 2 lnq
n = #usersm = #segmentsq = alphabet size
Capacity for the Combined Digit Model
The math• Look at one segment
• Define counters Σα= #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)
𝑰+-
Fθ
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 (2q-1)-dim. hypersphere.• Jacobian J
• Pay-off function Tr(JTJ)€
γw2 = Pr[W = w | Σ =σ ]
€
uα2 = pα
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, t1=1) RDM-like behaviour; weak dependence on t2
• 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 ...
• higer-dimensional space
• nasty constraint on the mapping
• Numerics for binary alphabet
• constrained geodesics in 2 dimensions
• graphs show how attack parameters (r, t1, t2) affect capacity
• useful for code design
• Future work (perhaps ...)
• change attack model to get analytic results
Questions?
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