Teoría de las tres dimensiones

This subsection explains probabilistic fingerprinting schemes with the example of the symmetric

Tardos scheme, consisting of the fingerprint generation process with a continuous distribution

function according to Tardos in [114] as well as an alternative fingerprint generation with a discrete distribution function. This is followed by the fingerprint tracing algorithm according to [121].

4.5.1 Tardos fingerprint generation

For the fingerprint generation process the distributor needs to choose the parameters"1 and c0

to calculate the minimum code length m the fingerprinting scheme mathematically proves to satisfy the chosen error probabilities. The code length formula is given as

m_{≥ dm}c₀2ln 1

"1

(4)

The code length parameter d_m depends on the chosen fingerprinting scheme. That is, for the original Tardos code [114] d_m is set as100. Recently it was shown, that in the asymptotic case

the parameter can be lowered to d_m ≥ π₂2, [64]. However, this is provable secure only for very large and therefore not realistic collusions, that are not considered within this thesis.

With it, the distributor generates an n× m matrix X, where the n denotes the number of users to be accommodated in his system. Ergo the jth row of the matrix corresponds to the fingerprint which is later embedded in the copy that is released to customer j ∈ {1, ..., n}. The entries Xji of matrix X are generated in two steps:

First, the distributor picks m independent random numbers{pi}mi=1 over the interval pi ∈ [t, 1 −

t], with a so called cutoff parameter t < 1₄. Each pi = sin2(ri) is selected by picking uniformly at random the value r_i ∈t,π₂ − t with 0 < t <π₄, wheresin2(t) = t. The choice for the cutoff parameter t in the original Tardos code [114] was t = (300c0)−1, without giving any rationale.

In [121] Škori´c et al. prove that this was suboptimal, the asymptotical optimal choice is t ← γ 4c −4/3 0 withγ = 2 3π
2/3

, as proven by [66]. More concerns regarding the cutoff parameter can be found in the appendix I or in [55] and [62].

Second, the matrix X is filled by picking each entry X_ji independently from the binary alphabet {0, 1} according to P[Xji = 1] = pi . The independence of the entries Xji holds only in the sec- ond step, since two overall random bits X_ji and X_j0iof the same column are positively correlated since both of them have a higher probability to be 1 if p_i is large.

The most famous distribution and also the most applied throughout this thesis is the arcsine

distribution introduced in [114] by Tardos. Its corresponding continuous probability density

function f is defined within the interval[t, 1 − t], with t close to zero. It is symmetric around 0.5 and heavily biased towards values close to the limits of[t, 1 − t], as can be seen in the left graphic of figure 2. This is motivated by the marking assumption explained in subsection 4.4. It is the only restriction on the colluders attack model and it is more likely to apply to these columns with a high bias. This choice of the distribution of p_i is used to show that the colluders’ attack model only has a minor effect on their chance to be caught. For the binary case the probability density function is defined as

Definition 18 (The Tardos probability density distribution function)

f(p) = 1

2 arcsin(1 − 2t) 1

p p(1 − p), p∈ [t, 1 − t]. (5)

4.5.2 Fingerprint generation with discrete bias distribution

This subsection describes an alternative fingerprint matrix generation by means of discrete ver- sions of the arcsine distribution mentioned in the afore subsection. This type of fingerprint generation will be relevant in chapters 7.3 and 7.4 and are used for instance in the approaches by [43], [84] and recently [65]. In the latter, Laarhoven and de Weger proved that for very large collusions (c0−→ ∞), the results for the discrete distribution tend to those for the continuous

distribution, what is adumbrated in figure 3.

Instead of selecting the m bias values pi as for the Tardos fingerprint generation, for the dis- crete generation these are selected according to (discrete) supporting points, represented by finite random variablesP , with Pr[P = p] = q = Pr[P = 1 − p]. Note that the value q is the

Figure 2: Tardos probability density function and score functionsg₁andg₀, according to [88] equivalent to the value f(p) that describes the occurrences of the probability p in the finger- print matrix X for the Tardos fingerprint generation. The number of those random variables is prescribed by the number of expected colluders c₀. Hence we get(p, q) ∈ P_c0.

According to Nuida et al. [85] the distributions are related to Gauss-Legendre distributions. For

T ≥ 1, let L_T(x) = 1 2T_T!  d d x ‹T (x2 − 1)T

be the T -th normalized Legendre polynomial with LT(1) = 1. This polynomial has T simple roots that can be taken as supporting points:

(p, q)T =  x+ 1 2 , 2 (1 − x2₎3/2_L0 T(x)2  for L_T(x) = 0 [85].

But as the polynomials L_T(x) get more and more complex for increasing T, a method to ap- proximate this distribution is presented by Laarhoven and de Weger in [65]:

Let x_1,T< ... < x_T,T denote the T roots of the polynomial L_T(x) and let p_k,T = (x_k,T+ 1)/2 and

q_k,T = 2/((1−x2)3/2L_T0(x)2) denote the corresponding values from the distribution (p_k,T, q_k,T) ∈

P2T. Laarhoven and de Weger show in [65] that pk,T = sin2 πk2T
+ o(1) and qk,T = 1/T + o(1/c)

for T → ∞.

Therefore, instead of computing the roots of the more and more complex polynomial L_T(x), one can approximate the values(p_k,T, q_k,T) by computing

ˆ p_k,T = sin2 _4k − 1 8T+ 4π ‹ and ˆq_k,T = 1 T

for1≤ k ≤ T . This gives the following fingerprint generation process:

1. Initialization: Let m= dmc0ln(1/"1) denote the code length and set T = dc0/2e. First

select m bias values p_i, for i= 1, ..., m, by drawing k for each value uniformly at random from{1, ..., T } and setting pi= sin2

4k−1 8T+4π
.

5 10 15 20 25 30 0 2 4 6 8 10 12 Colluders

Code Length Parameter d

m

Continuous arcsine−distribution

Discrete distribution by Laarhoven and de Weger Asymptotic code length

Figure 3: Approximated code length parameterd_mfor the continuous arcsine distribution [114] and the discrete distribution by [65] with the symmetric score function [121], taken from [88]

2. Fingerprint generation: Each entry X_j,i of the fingerprint matrix is selected indepen-

dently from the binary alphabet withPr[X_j,i= 1] = pi.

Figure 3 shows the code length parameter for the continuous arcsine-distribution and for the discrete distribution by Laarhoven and de Weger [65]. In both cases, the tracing algorithm from

Škori´c et al. [121] (see below) is assumed.

4.5.3 Symmetric Tardos tracing algorithm

When the distributor receives an unauthorized copy and the fingerprint detection process out- puts the potentially manipulated fingerprint y, he initializes the fingerprint tracing algorithm. In case he chooses the symmetric Tardos tracing algorithm by Škori´c et al. [121], he first creates the single accusation score S_j between each users’ fingerprint X_j and the attacked fingerprint y. The corresponding single accusation score according to (12) is defined as follows.

Definition 19 (Symmetric Tardos accusation score) For an arbitrary user j, with accusation

functions g₁ and g₀defined as

g₁(p) := v t1− p p g0(p) := − v t p 1− p,

the symmetric Tardos accusation score is calculated as

S_j= m X i=1 ϕ yi; Xj(i)
= m X i=1 δyi,Xjig1(p (i) yi) + [1 − δyi,Xji]g0(p (i) yi). (6)

Figure 4: Normalized distribution of the accusation scores in [121]: innocents’ scores (left), col- luders’ scores (right).

where δ_yi,Xji denotes the Kronecker Delta and p(i)_y

i stands for the probability of the symbol in yi

within the entries of the fingerprint matrixX .

The accusation score functions are depicted in figure 2.

To determine, whether a fingerprint is suspicious or not a threshold decision is applied. This means, the distributor suspects a user j to have partaken in the collusion, if the corresponding accusation score Sj exceeds a predefined threshold Z,

∀ j ∈ {1, ..., n} : Sj> Z ⇒ j ∈ T,

where T denotes the output set of the tracing algorithm, i.e. the set of fingerprints that are suspected of having partaken in the collusion.

The threshold Z depends on the chosen maximum number of colluders c₀ and of the chosen rate for"1,

Z ≤ dzc0ln

1

"1

, (7)

with a threshold parameter dz depending on the chosen fingerprint tracing algorithm. For the symmetric Tardos tracing algorithm, d_z is set asσ · π. Note that the standard deviation of the expected accusation scores of innocent users σ is assumed to be 1 for the symmetric Tardos scheme [121] as well as for the original Tardos [114] and most of its derivatives e.g. [86, 21, 73]. Figure 4 depicts the score distribution for the symmetric Tardos scheme [121]. Note that the score distributions and threshold Z have been normalized. The figure also illustrates the understanding of the error bounds"₁and"₂.