### Abstract

This paper investigates the relation between three different notions of privacy: identifiability, differential privacy, and mutual-information privacy. Under a unified privacy-distortion framework, where the distortion is defined to be the expected Hamming distance between the input and output databases, we establish some fundamental connections between these three privacy notions. Given a maximum allowable distortion D , we define the privacy-distortion functions \epsilon mathrm{ i}(D) epsilon d (D), and epsilon m(D) to be the smallest (most private/best) identifiability level, differential privacy level, and mutual information between the input and the output, respectively. We characterize \epsilon mathrm{ i}(D) and \epsilon mathrm{ d}(D),and prove that \epsilon i (D)- X\le epsilon mathrm d(D)\le \epsilon mathrm{ i}(D) for D within certain range, where epsilon X is a constant determined by the prior distribution of the original database X and diminishes to zero when X is uniformly distributed. Furthermore, we show that \epsilon mathrm{ i}}(D) and \epsilon mathrm{ m}(D) can be achieved by the same mechanism for D within certain range, i.e., there is a mechanism that simultaneously minimizes the identifiability level and achieves the best mutual-information privacy. Based on these two connections, we prove that this mutual-information optimal mechanism satisfies \epsilon -differential privacy with \epsilon mathrm{ d}(D)\le \epsilon \le \epsilon mathrm{ d}(D)+2\epsilon {X}. The results in this paper reveal some consistency between two worst case notions of privacy, namely, identifiability and differential privacy, and an average notion of privacy, mutual-information privacy.

Original language | English (US) |
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Article number | 7498650 |

Pages (from-to) | 5018-5029 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 62 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2016 |

### Keywords

- Differential privacy
- Hamming distance
- identifiability
- mutual information
- rate-distortion

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

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## Cite this

*IEEE Transactions on Information Theory*,

*62*(9), 5018-5029. [7498650]. https://doi.org/10.1109/TIT.2016.2584610