Accusation probabilities
in Tardos codes
Antonino Simone and Boris Škorić
Eindhoven University of Technology
WISSec 2010, Nov 2010
Outline

Introduction to forensic watermarking
◦ Collusion attacks
◦ Aim

Tardos scheme
◦ q-ary version
◦ Properties

Performance of the Tardos scheme
◦ False accusation probability

Results & Summary
Forensic Watermarking
original
content
original
content
WM secrets
payload
Embedder
content with
hidden payload
payload
Detector
WM secrets
ATTACK
Payload = some secret code indentifying the recipient
Collusion attacks
"Coalition of pirates"
 = "detectable positions"
pirate #1
1
1
1
0
1
0
1
0
0
0
0
1
#2
1
0
1
0
1
0
1
0
1
0
1
1
#3
1
0
1
0
1
0
1
0
0
0
1
1
#4
1
1
1
0
0
0
1
1
0
0
0
1
1
0/1
1
0
0/1
0
1
0
0/1
1
Attacked
Content
0/1 0/1
Aim
Trace at least one pirate from detected
watermark
BUT
Resist large coalition
longer code
Low probability of innocent accusation (FP) (critical!)
longer code
Low probability of missing all pirates (FN) (not critical)
longer code
AND
Limited bandwidth available for
watermarking code
q-ary Tardos scheme (2008)
m content segments
biases
Symbol biases
drawn from
distribution F
embedded
symbols
• Arbitrary alphabet size q
• Dirichlet distribution F
n users
c pirates
watermark
after attack
Symbols allowed
p1A
p1B
p1C
p2A
p2B
p2C
piA
piB
piC
pmA
pmB
pmC
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
C
A
B
A
A
B
C
=y
Tardos scheme continued
Accusation:
• Every user gets a score
• User is accused if score > threshold
• Sum of scores per content segment
• Given that pirates have y in segment i:
• Symbol-symmetric
Properties of the Tardos scheme

Asymptotically optimal
◦ m c2 for large coalitions, for every q
◦ Previously best m c4
◦ Proven: power ≥ 2
Random code book
 No framing

◦ No risk to accuse innocent users if coalition
is larger than anticipated

F, g0 and g1 chosen ‘ad hoc’ (can still be
improved)
Accusation probabilities
Result: majority
voting minimizes u
m = code length
c = #pirates
Pirates want to
minimize u and
make longer the
innocent tail
threshold
u = avg guilty score
Curve shapes
depend on:
 F, g0, g1 (fixed ‘a
priori’)
 Code length
 # pirates
 Pirate strategy
guilty
innocent
u
total score (scaled)
Central Limit Theorem  asymptotically Gaussian shape (how fast?)
2003  2010: innocent accusation curve shape unknown… till now!
Approach
Steps:
1. S = i Si
Si
= pdf of total score S
S
= InverseFourier[
2.
3.
4.
5.
Fourier transform property:
]
Compute
• Depends on strategy
• New parameterization for attack strategy
Compute
•
•
•
Taylor
Taylor
Taylor
Main result: false accusation
probability curve
threshold/√m
Example:
exact FP
majority voting attack
log10FP
FP is
Result
from
Gaussian
70 times less than Gaussian approx in this example
But
Code 2-5% shorter than predicted by Gaussian approx
Summary
Results:
 introduced a new parameterization of the attack strategy
 majority voting minimizes u
 first to compute the innocent score pdf
◦ quantified how close FP probability is to Gaussian
◦ sometimes better then Gaussian!
◦ safe to use Gaussian approx
Future work:
 study more general attacks
 different parameter choices
Thank you for your attention!