Asymptotic fingerprinting capacity in the Combined Digit Model

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24 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Asymptotic fingerprinting capacity
in the

Combined Digit Model


Dion Boesten and Boris
Š
kori
ć


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

Embedder

Detector

original

content

unique

watermark

watermarked

content

unique

watermark

original

content

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

a
llowed

symbols

A

C

A

B

A

A

B

C

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

|ψ|

Noise parameter r. Mixing parameters t
1

≥ t
2

≥ t
3

...

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

𝑝
1

𝑝
1

𝑝
1

𝑝
2

𝑝
2

𝑝
2

𝑝
𝑖

𝑝
𝑖

𝑝
𝑖

𝑝
𝑚

𝑝
𝑚

𝑝
𝑚

symbol

biases

𝑚

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[X
ij

= α] = p

.


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:


C
2

1
2
c
2
ln
2

C
q

q

1
2
c
2
ln
q
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
)

𝑰

+

-

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 c
p


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)



w
2

Pr[
W

w
|



]

u

2

p

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?