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Quantifying the Protection Level of a Noise Candidate for Noise Multiplication Masking Scheme

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Privacy in Statistical Databases (PSD 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11126))

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Abstract

When multiplicative noises are used to perturb a set of original data, the data provider needs to ensure that the original values are not likely to be learned by data intruders from the noise-multiplied data. Different attacking strategies for unveiling the original values have been recognised in the literature, and the data provider needs to ensure that the noise-multiplied data is protected against these attacking strategies by selecting an appropriate noise generating variable. However, there are many potential attacking strategies, which makes the quantification of the protection level of a noise candidate difficult. In this paper, we argue that, to quantify the protection level a noise candidate offers to the original data against an attacking strategy, the data provider might look at the average value disclosure risk it produces. Correspondingly, we propose an optimal estimator which maximizes the average value disclosure risk. As a result, the data provider could use the maximized average value disclosure risk as a single measure for quantifying the protection level a noise candidate offers to the original data. The measure could help the data provider with the process of noise generating variable selection in practice.

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Acknowledgements

We thank the anonymous reviewers for their constructive comments on the paper. This research has been conducted with the support of the Australian Government Research Training Program Scholarship.

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Authors and Affiliations

  1. National Institute for Applied Statistics Research Australia, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW, 2500, Australia

    Yue Ma, Yan-Xia Lin, Pavel N. Krivitsky & Bradley Wakefield

Authors
  1. Yue Ma
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  2. Yan-Xia Lin
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  3. Pavel N. Krivitsky
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  4. Bradley Wakefield
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Corresponding author

Correspondence to Yue Ma .

Editor information

Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Josep Domingo-Ferrer

  2. University of Valencia, Burjassot, Spain

    Francisco Montes

Appendix

Appendix

In this section we show how \(z_i^{opt}\) is derived. Suppose for a set of original data \(\{y_i\}_{i=1}^n\), the following probabilistic disclosure risk measure to be used by the data provider.

$$\begin{aligned} P(\frac{|\tilde{Y}_i-Y_i|}{Y_i}<\delta )=P(\frac{|\tilde{Y}-Y|}{Y}<\delta ), i=1,\cdots , n \end{aligned}$$

In the following we assume \(Y>0\), \(C>0\), \(\tilde{Y}=g(Y^*)\), where \(Y^*=CY\). We observe that the disclosure risk of an observation y is given as:

$$\begin{aligned} P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y=y)=\int _{0}^{\infty }I(\frac{g(y^*)}{1+\delta }<y<\frac{g(y^*)}{1-\delta })f_{Y^*|Y}(y^*|Y=y)dy^* \end{aligned}$$

Therefore \(P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y)\) is a function of random variable Y.

The average disclosure risk for the original data is

$$\begin{aligned} R_{overall}=\frac{\sum _{i=1}^n P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y=y_i)}{n} \end{aligned}$$

Suppose \(E_Y(P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y))\) exists, therefore we have

$$\begin{aligned} R_{overall}\overset{P}{\rightarrow }E_Y(P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y)) \end{aligned}$$

as \(n\rightarrow \infty \).

The objective is to find an expression of \(g(Y^*)\) which maximizes \(R_{overall}\) as \(n\rightarrow \infty \). Because \(\{Y^*|Y=y\}=yC\), therefore \(f_{Y^*|Y}(y^*|Y=y)=\frac{1}{y}f_C(\frac{y^*}{y})\). We observe the following:

$$\begin{aligned} \begin{aligned} E_Y(P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y))&=\int _{0}^{\infty }\int _{0}^{\infty } I(\frac{g(y^*)}{1+\delta }<y<\frac{g(y^*)}{1-\delta })f_{Y^*|Y=y}(y^*)dy^* f_Y(y)dy\\&=\int _{0}^{\infty }\int _{0}^{\infty } I(\frac{g(y^*)}{1+\delta }<y<\frac{g(y^*)}{1-\delta })\frac{1}{y} f_C(\frac{y^*}{y})f_Y(y)dydy^*\\&=\int _{0}^{\infty }f_{Y^*}(y^*)\int _{0}^{\infty } I(\frac{g(y^*)}{1+\delta }<y<\frac{g(y^*)}{1-\delta })f_{Y|Y^*=y^*}(y)dydy^*\\&=\int _{0}^{\infty }f_{Y^*}(y^*)\int _{\frac{g(y^*)}{(1+\delta )}}^{\frac{g(y^*)}{(1-\delta )}} f_{Y|Y^*=y^*}(y)dydy^*\end{aligned}\end{aligned}$$
(2)

Therefore, \(E_Y(P(\frac{|g(Y^*)-Y|}{Y}<\delta |Y))\) is maximized if \(\int _{\frac{g(y^*)}{(1+\delta )}}^{\frac{g(y^*)}{(1-\delta )}} f_{Y|Y^*=y^*}(y)dy\) is maximized. The form of \(g(y^*)\) which maximizes \(\int _{\frac{g(y^*)}{(1+\delta )}}^{\frac{g(y^*)}{(1-\delta )}} f_{Y|Y^*=y^*}(y)dy\) is \(z_{opt}=argmax_{g(y^*)} \int _{\frac{g(y^*)}{(1+\delta )}}^{\frac{g(y^*)}{(1-\delta )}} f_{Y|Y^*=y^*}(y)dy\). Therefore, the optimal estimator \(Z_{opt}\) takes the following form

$$\begin{aligned} Z_{opt}=argmax_{g(Y^*)} \int _{\frac{g(Y^*)}{(1+\delta )}}^{\frac{g(Y^*)}{(1-\delta )}} f_{Y|Y^*}(y)dy \end{aligned}$$

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Ma, Y., Lin, YX., Krivitsky, P.N., Wakefield, B. (2018). Quantifying the Protection Level of a Noise Candidate for Noise Multiplication Masking Scheme. In: Domingo-Ferrer, J., Montes, F. (eds) Privacy in Statistical Databases. PSD 2018. Lecture Notes in Computer Science(), vol 11126. Springer, Cham. https://doi.org/10.1007/978-3-319-99771-1_19

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  • DOI: https://doi.org/10.1007/978-3-319-99771-1_19

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