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Impact of data correlation on privacy budget allocation in continuous publication of location statistics

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Abstract

Continuous publication of statistics collected from various location-based applications may compromise users’ privacy as the statistics could be procured from users’ private data. Differential Privacy (DP) is a new privacy notion that offers a strong privacy guarantee to all users who participate in the statistics. However, the existing DP mechanism for continuous publication of location statistics provides a privacy guarantee with the assumption that the data-points of users stream at consecutive timestamps are independent. In reality, users’ data-points may be temporally correlated, resulting in more privacy leakage due to an inadequate supply of privacy budget to the timestamps where the data-points are correlated. In this paper, we present a reformulated differential privacy definition to quantify the impact of temporal correlation on privacy leakage. Then, we introduce a privacy budget allocation method for allocating an adequate amount of privacy budget to each successive timestamps under the protection of differential privacy. Our solution adopts w-event privacy for continuously releasing statistics over infinite streams. The main idea is to check the dissimilarity between statistics at each timestamp and decide whether to publish current statistics or last release statistics. Finally, we evaluate the data utility of our proposed method by presenting experimental results for real and synthetic data sets.

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Authors and Affiliations

  1. National Institute of Technology Warangal, Hanamkonda, India

    D. Hemkumar, S. Ravichandra & D. V. L. N. Somayajulu

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  1. D. Hemkumar
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  2. S. Ravichandra
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  3. D. V. L. N. Somayajulu
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Correspondence to D. Hemkumar.

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This article belongs to the Topical Collection: Special Issue on Privacy-Preserving Computing

Guest Editors: Kaiping Xue, Zhe Liu, Haojin Zhu, Miao Pan and David S.L. Wei

Appendix:

Appendix:

Theorem 3

Prove that

$${\sum}_{j=k-w+1}^{k}(\epsilon/4 - [{\Sigma}_{j=k-w+1}^{k-1}\epsilon_{j}])/2 \leq \epsilon/4$$

.

Proof

Sub-mechanism \({\mathscr{M}}^{2}\) allocates a privacy budgets per timestamps in exponential decreasing fashion i.e., (𝜖/8, 𝜖/16, 𝜖/32,...). Then LHS is,

$$ {\sum}_{j=k-w+1}^{k}(\epsilon/4 - {\Sigma}_{j=k-w+1}^{k-1}\epsilon_{j})/2 = (\frac{\epsilon}{8} + \frac{\epsilon}{16} + \cdot\cdot\cdot + \frac{\epsilon}{2^{n+2}}) $$
$$ = {\Sigma}_{i=3}^{n}\frac{\epsilon}{2^{i}} \leq \epsilon/4 $$

We can rewrite

$$ {\Sigma}_{i=3}^{n}\frac{\epsilon}{2^{i}} \leq \epsilon/4 \text{into} {\Sigma}_{i=1}^{n}\frac{\epsilon}{2^{i}} \leq \epsilon $$

Let assume that 𝜖 = 1

$$ {\Sigma}_{i=1}^{n}\frac{1}{2^{i}} = {\Sigma}_{i=1}^{1}(\frac{1}{2})(\frac{1}{2})^{i-1} = \frac{1}{2}\frac{(1-(\frac{1}{2})^{i})}{(1-(\frac{1}{2}))} $$

By mathematical induction: Basis: n = 1

$$ {\Sigma}_{i=1}^{1}(\frac{1}{2})(\frac{1}{2})^{1-1} = \frac{1}{2}\frac{(1-(\frac{1}{2})^{1})}{(1-(\frac{1}{2}))} \leq 1 $$

Inductive step: Assume true for n = p

$$ {\Sigma}_{i=1}^{p}(\frac{1}{2})(\frac{1}{2})^{0}+(\frac{1}{2})(\frac{1}{2})^{1}+\cdot\cdot\cdot+(\frac{1}{2})(\frac{1}{2})^{p-1} $$
$$ = \frac{1}{2}\frac{(1-(\frac{1}{2})^{p})}{(1-(\frac{1}{2}))} \leq 1 $$

Prove true for n = p + 1

$$ {\Sigma}_{i=1}^{p+1}(\frac{1}{2})(\frac{1}{2})^{0}+(\frac{1}{2})(\frac{1}{2})^{1}+\cdot\cdot\cdot+(\frac{1}{2})(\frac{1}{2})^{p-1}+ (\frac{1}{2})(\frac{1}{2})^{p} $$
$$ = \frac{1}{2}\frac{(1-(\frac{1}{2})^{p+1})}{(1-(\frac{1}{2}))} $$
$$ = \frac{1}{2}\frac{(1-(\frac{1}{2})^{p})}{(1-(\frac{1}{2}))}+(\frac{1}{2})(\frac{1}{2})^{p} $$
$$ = \frac{1}{2}\frac{(1-(\frac{1}{2})^{p}+(\frac{1}{2})^{p}-(\frac{1}{2})^{p+1})}{(1-(\frac{1}{2}))} $$
$$ = \frac{1}{2}\frac{(1-(\frac{1}{2})^{p+1})}{(1-(\frac{1}{2}))}=\frac{1}{2}\frac{(1-(\frac{1}{2})^{p+1})}{(1-(\frac{1}{2}))} \leq 1 $$

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Hemkumar, D., Ravichandra, S. & Somayajulu, D.V.L.N. Impact of data correlation on privacy budget allocation in continuous publication of location statistics. Peer-to-Peer Netw. Appl. 14, 1650–1665 (2021). https://doi.org/10.1007/s12083-021-01078-6

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