Tradeoff Between the Price of Distributing a Database and Its Collusion Resistance Based on Concatenated Codes

Abstract

The purchasing of customer databases, which is becoming more and more common, has led to a big problem: the use of purchased databases to mount a collusion attack, which is when purchasers of a database illegally combine their versions of it in order to de-anonymize the private information it contains. However, the purchasing of customer database is only available in the black market. In this paper, we first investigated the relationship between the price of distributing a database and its collusion resistance. A fingerprint is embedded in database so that illegal distributors can be identified. The fingerprints are constructed on the basic of concatenated codes. After the fingerprint is embedded, the price of distributing the database and its collusion resistance are modelled as decreasing functions. The less expensive the database is, the less collusion resistance the database owner deals with. There are upper and lower bounds for the collusion capabilities. To the best of our knowledge, this scheme is unique in that the tradeoff between the price of distributing a database and its collusion resistance is based on a mathematical model. Second, we propose a guideline to sell customer database legally with profit and risk evaluation.

A Omitted Proofs from Sect. 5

Proof of Theorem 19

Proof

Set $n = n_1 n_2$. Since C is a matrix of $n_1n_2 \times 2^{k_1k_2}$, each codeword of C has a length of $n_1n_2$. Each database is represented by a codeword of C. Since the Hamming weight in a codeword is $n_1w$, each entry of a codeword is randomly assigned to 1 with probability $p_j = \frac{n_1 w_j}{n_1 n_2} = \frac{w_j}{n_2}$.

Colluders achieve perfect collusion if $\prod _{i \in C, i=1}^{|C|} x_{ij}=0$ for all $j=1,\ldots ,n$. The probability of a row contains at least one 0 is:

$$\begin{aligned} 1 - \prod _{i=1}^c p_{j_i} = 1 - \frac{\prod _{i=1}^{c}w_{j_i}}{n_2^c} \le 1 - \left( \frac{w_{min}}{n_2} \right) ^c \end{aligned}$$

(3)

The probability of n rows whose each row containing at least one 0 each is:

$$\begin{aligned} \left( 1 - \prod _{i=1}^c p_{j_i} \right) ^n \le \left( 1 - \left( \frac{w_{min}}{n_2} \right) ^c \right) ^{n_1 n_2} \end{aligned}$$

(4)

When $C_{in}$ is a constant codeword weight code, $w_{min} = w_{max} = w_i$. Therefore, the Eq. 4 holds.

Proof of Theorem 22

Proof

In accordance with Definition 12, the price of distributing a database is

$$\begin{aligned} \frac{1}{wt(\texttt {a~codeword}) + 1}< & {} price(\texttt {a~database})= \frac{1}{wt(\texttt {a~codeword}) + \frac{sum(\texttt {a~codeword}) - wt(\texttt {a~codeword})}{sum(\texttt {a~codeword})}} \nonumber \\< & {} \frac{1}{w_{min}} \end{aligned}$$

(5)

Since C has $2^{k_1k_2}$ codewords, the total price of distributing databases embedded with these codewords is

$$\begin{aligned} \sum _{i=1}^{2^{k_1 k_2}} \frac{1}{wt(\texttt {a~codeword}) + \frac{sum(\texttt {a~codeword}) - wt(\texttt {a~codeword})}{sum(\texttt {a~codeword})}} < 2^{k_1 k_2} \times \frac{1}{n_1w_{min}} \end{aligned}$$

Since the number of attributes of the database is unchanged, the block length of $C_{in}$ is unchanged. Suppose that there is another $w'_{max}$-weight code $C'_{in}$ $[n_2, k'_2, d'_2]_2$. The price of distributing the database when using $C'$ generated using $C_{out}$ and $C'_{in}$ is:

$$\begin{aligned} \sum _{i=1}^{2^{k_1 k'_2}} \frac{1}{wt(\texttt {a~codeword}) + \frac{sum(\texttt {a~codeword}) - wt(\texttt {a~codeword})}{sum(\texttt {a~codeword})}} > 2^{k_1 k'_2} \times \frac{1}{n_1 w'_{max} + 1} \end{aligned}$$

To complete our proof, we prove that if the price of distributing the databases when using $C'$ is larger than the price of distributing the databases when using C, $k_2 > k'_2$. Indeed, we have:

$$\begin{aligned} 2^{k_1 k_2} \times \frac{1}{n_1w_{min}}> & {} \sum _{i=1}^{2^{k_1 k_2}} \frac{1}{wt(\texttt {a~codeword}) + \frac{sum(\texttt {a~codeword}) - wt(\texttt {a~codeword})}{sum(\texttt {a~codeword})}}\end{aligned}$$

(6)

$$\begin{aligned}> & {} \sum _{i=1}^{2^{k_1 k'_2}} \frac{1}{wt(\texttt {a~codeword}) + \frac{sum(\texttt {a~codeword}) - wt(\texttt {a~codeword})}{sum(\texttt {a~codeword})}}\end{aligned}$$

(7)

$$\begin{aligned}> & {} 2^{k_1 k'_2} \times \frac{1}{n_1 w'_{max} + 1}\end{aligned}$$

(8)

$$\begin{aligned} \Rightarrow 2^{k_1 (k_2 - k'_2)}> & {} \frac{n_1 w_{min}}{n_1 w'_{max} + 1} \end{aligned}$$

(9)

We consider three cases:

• 1. If $k_2 < k'_2$,

  $$\begin{aligned} 2^{k_1 (k_2 - k'_2)} \le \frac{1}{2^{k_1}} < \frac{1}{n_1} < \frac{n_1 w_{min}}{n_1 w'_{max} + 1} \end{aligned}$$

  (10)

  because $0 < w_{min}, w'_{max} \le n_2 \le n_1$. It is contradicted to inequality 6.

• 2. If $k_2 = k'_2$, $d_2 = d'_2$ because $k_2$ and $k'_2$ are the largest numbers such that $k_2 \le n_2 - d_2 + 1$ and $k'_2 \le n_2 \ d'_2 + 1$. Therefore, the price of distributing the databases when using $C'$ is equal to the price of distributing the databases when using C. It is contradicted to our hypothesis.

• 3. If $k_2 > k'_2$, the inequality 6 holds.

Thus, if the price of distributing the databases when using $C'$ is larger than the price of distributing the databases when using C, $k_2 > k'_2$. If $k'_2 < k_2$, $d'_2 > d_2$.

According to Corollary 20, if the relative distance of $C_{in}$ increases (decreases), the probability of perfect collusion decreases (increases). According to Corollary 15, if the code rate of $C_{in}$ increases, the price of distributing databases using C increases. Therefore, the lower the price of distributing a database, the less collusion resistance the database owner deals with.