Collusion-resistant Collusion-resistant fingerprinting for fingerprinting for

multimediamultimedia

Collusion-resistant Collusion-resistant fingerprinting for fingerprinting for

multimediamultimediaWade Trappe, Min Wu, Wade Trappe, Min Wu,

K.J. Ray LiuK.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 symbol

s {0,1} to {-1,1} via for use in CDMA-based watermarking.• We assume that when a sequence of watermark

s is averaged, the effect it has is that the resulting binary is the logical AND of the codewords

• For example ,when the codes (1110) and (1101) are combined, the result is (1100).

jc

12)( xxf

12)( xxf

ACC (2)• Definition 1. A binary code such that the logical AN

D 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 of these vectors are combined under AND, that this combination is unique .

},,{ 1 nccC

1nk

ACC -BIBD (1)• Definition 2: A 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.

),( ),,( kv

ACC –BIBD (2)• A -(BIBD) has blocks. Correspo

nding to a block design is the incidence matrix defined by

• If we assign the codewords as the bit-complement of the column vectors of then we have a (k-1)-resilient ACC.

)/()( 22 kkvvb ),,( kvbv

)( ijmM

otherwise 0

blockth the tobelongselement th theif 1 jimij

M

ACC –BIBD (3)• Theorem 1. Let be a (v,k,1)-BIBD, and the corre

sponding incidence matrix. If the codevectors are assigned as the bit complement of the columns of ,then the resulting scheme is a (k-1)-collusion resistant code.

),( M

M

ACC – BIBD (4)• We now present an example. The

following is the bit-complement of the incidence matrix for a (7,3,1)-BIBD:

1001011

0101101

0110011

0011110

1010101

1100110

1111000

M

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 is mapped to by .

• The CDMA watermark is then • When two watermarks are averaged, the

locations where the corresponding ACC codes agree and have a value of 1 identify the colluding users.

M }1{12)( xxf

jj scf

v

1j)(w

ACC –BIBD (6)• For example, let

be the watermarks for the first two columns of the above (7,3,1) code, then

has coefficient vector (-1,0,0,0,1,0,1).

76543212

76543211

w

w

sssssss

sssssss

2/)w(w 21

Restriction of BIBD• In general, do not necessarily exist for an

arbitrary choice of v, and k.• The condition that b must be an integer restricts so

me possibilities for v, and k, and for a given triple there may not exist a .

• The Bose construction builds Steiner triple systems when ,and the Skolem construction builds Steniner triple systems when . Additionally, can be constructed when p is of prime power.

BIBDs),,( kv

)6(mod 3v

BIBD),,( kv

)6(mod 1v

BIBD)1,,( ppd

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 ,

where the matrices on the diagonal correspond to matrices for smaller codes.

),,,( 21 sCCCdiagC jC

Simulation (1)• In order to demonstrate the capabilities of using a

n ACC code with CDMA watermarking to fingerprint users and detect colluders, we used an additive spread spectrum watermarking scheme, where the perceptually weighted watermark was added to block DCT coefficients.

• 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 given by the binary operation .

• 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)

5Z)5)(mod33( yxyx

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 different spreading sequences used when user 1 and user6 average their differently marked images and then compress using JPEG with quality factor 50%.

Simulation (4)• We present a histogram containing

the Z statistics form roughly 200 pairs of users in figure 2 when the watermarked images are compressed using JPEG with a quality factor of 50%.