# Collusion-resistant fingerprinting for multimedia by gegeshandong

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• pg 1
```									Collusion-resistant
fingerprinting for
multimedia
K.J. Ray Liu
Anti-collusion codes
• Anti-collusion codes (ACC) are
designed to be resistant to averaging,
and able to exactly identify groups of
colluders.
ACC (1)
•    We shall describe codes using the binary
symbols {0,1} to {-1,1} via f ( x)  2 x  1
for use in CDMA-based watermarking.
f ( x)  2 x  1

•    We assume that when a sequence of
watermarks is averaged, the effect it has is
that the resulting binary is the logical AND of
j
the codewords c
•    For example ,when the codes (1110) and (1101)
are combined, the result is (1100).
ACC (2)
• Definition 1.
A binary code C  {c ,  , c } such that the
1         n

logical AND of any subset of k or fewer
codevectors in non-zero and distinct from the
logical AND of any other subset of k or fewer
codevectors is a k-resilient anti-collusion code,
or an ACC code.
• For example, when n=4,C={1110,1101,1011,0111}.
It is easy to see when k  n 1 of these
vectors are combined under AND, that this
combination is unique .
ACC -BIBD (1)
• Definition 2:
A (v, k ,  ) balanced incomplete block

design (BIBD) is a pair (  , ) ,where  is a
collection of k-element subsets (blocks) of a
v-element set ,such that each pair of

elements of occur together in exactly 
blocks.
ACC –BIBD (2)
• A (v, k ,  ) -(BIBD) has b   (v  v) /(k  k ) blocks.
2        2

Corresponding to a block design is the v  b
incidence matrix M  (mij ) defined by

1 if the ith element belongs to the jth block
mij  
0                                  otherwise
• If we assign the codewords as the bit-
complement of the column vectors of M then we
have a (k-1)-resilient ACC.
ACC –BIBD (3)
• Theorem 1.
Let (  , ) be a (v,k,1)-BIBD, and M the
corresponding incidence matrix. If the
codevectors are assigned as the bit
complement of the columns of M ,then
the resulting scheme is a (k-1)-collusion
resistant code.
ACC – BIBD (4)
• We now present an example. The
following is the bit-complement of the
incidence matrix for a (7,3,1)-BIBD:
0   0   0   1   1   1   1
                         
0   1   1   0   0   1   1
1   0   1   0   1   0   1
                         
M  0   1   1   1   1   0   0
1   1   0   0   1   1   0
                         
1   0   1   1   0   1   0
                         
1   1   0   1   0   0   1
ACC – BIBD (5)
• This code requires 7 bits for 7 users and provides
2-resiliency since any two column vectors share a
unique pair of 1 bits.
• Each column vector c of M is mapped to {1}
by f ( x)  2 x  1 .

v
• The CDMA watermark is then w                f (c j )s j
j1
• When two watermarks are averaged, the locations
where the corresponding ACC codes agree and
have a value of 1 identify the colluding users.
ACC –BIBD (6)
• For example, let
w1   s1  s2  s3  s4  s5  s6  s7
w 2   s1  s2  s3  s4  s5  s6  s7
be the watermarks for the first two
columns of the above (7,3,1) code, then
(w  w ) / 2 has coefficient vector
1    2

(-1,0,0,0,1,0,1).
Restriction of BIBD
• In general, (v, k ,  )  BIBDs do not necessarily exist
for an arbitrary choice of v, and k.
• The condition that b must be an integer restricts
some possibilities for v, and k, and for a given
triple there may not exist a (v, k ,  )  BIBD    .
• The Bose construction builds Steiner triple systems
when v  1 (mod 6) ,and the Skolem construction
builds Steniner triple systems when v  3 (mod 6) .
Additionally,( p d , p,1)  BIBD can be constructed
when p is of prime power.
Subgroup-based
constructions
• In many application, a group of users will be
suspected of likely colluding, but not with others.
This could be due to geographical of social
variables, or based upon a pervious precedent.
• Suppose the code matrix C is constructed as a
block diagonal matrix C  diag (C1 , C2 ,  , C s ) ,
where the matrices C j on the diagonal
correspond to matrices for smaller codes.
Simulation (1)
• In order to demonstrate the capabilities of using
an ACC code with CDMA watermarking to
fingerprint users and detect colluders, we used an
where the perceptually weighted watermark was
• The detection of the watermark is performed
without the knowledge of the host image via the Z
detection statistic.
• We used the 512*512 lenna as the host image for
the fingerprints.
Simulation(2)
• We assigned the code vectors as the column vectors
of the bit complement of the incidence matrix for a
(15,3,1) –BIBD that was constructed using the Bose
method with a symmetric idempotent quasigroup
structure on Z 5 given by the binary
operation x  y  (3x  3 y)(mod5)        .
• Two example code vectors that were assigned to
user 1 and 6 are
User 1 : (-1,-1,-1, 1, 1,1,1,1,1,1, 1,1,1,1,1)
User 6 : (-1,1, 1, -1, 1,1,1,1,1,1,-1,1,1,1,1)
Average:(-1, 0, 0, 0, 1,1,1,1,1,1, 0,1,1,1,1)
Simulation (3)
• An example of the behavior of the Z
statistic when two users collude is
depicted in figure 1, where we depict the
values of the Z statistic for the 15