## Abstract

This paper proves that an "old dog", namely - the classical Johnson-Linden Strauss transform, "performs new tricks" - it gives a novel way of preserving differential privacy. We show that if we take two databases, D and D′, such that (i) D′-D is a rank-1 matrix of bounded norm and (ii) all singular values of D and D′ are sufficiently large, then multiplying either D or D′ with a vector of iid normal Gaussians yields two statistically close distributions in the sense of differential privacy. Furthermore, a small, deterministic and public alteration of the input is enough to assert that all singular values of D are large. We apply the Johnson-Linden Strauss transform to the task of approximating cut-queries: the number of edges crossing a (S, \bar S)-cut in a graph. We show that the JL transform allows us to publish a sanitized graph that preserves edge differential privacy (where two graphs are neighbors if they differ on a single edge) while adding only O(|S|/∈) random noise to any given query (w.h.p). Comparing the additive noise of our algorithm to existing algorithms for answering cut-queries in a differentially private manner, we outperform all others on small cuts (|S| = o(n)). We also apply our technique to the task of estimating the variance of a given matrix in any given direction. The JL transform allows us to publish a sanitized covariance matrix that preserves differential privacy w.r.t bounded changes (each row in the matrix can change by at most a norm-1 vector) while adding random noise of magnitude independent of the size of the matrix (w.h.p). In contrast, existing algorithms introduce an error which depends on the matrix dimensions.

Original language | English |
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Article number | 6375319 |

Pages (from-to) | 410-419 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - 2012 |

Externally published | Yes |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: 20 Oct 2012 → 23 Oct 2012 |

## Keywords

- Differential privacy
- Graph cuts