**Strong Hardness of Privacy from Weak Traitor Tracing**

*Lucas Kowalczyk and Tal Malkin and Jonathan Ullman and Mark Zhandry*

**Abstract: **A central problem in differential privacy is to accurately answer a large family $Q$
of statistical queries over a data universe $X$. A statistical query
on a dataset $D \in X^n$ asks ``what fraction of the elements of $D$ satisfy a given
predicate $p$ on $X$?'' Ignoring computational constraints, it is possible to accurately
answer exponentially many queries on an exponential size universe while satisfying
differential privacy (Blum et al., STOC'08). Dwork et al. (STOC'09) and Boneh and
Zhandry (CRYPTO'14) showed that if both $Q$ and $X$ are of polynomial size,
then there is an efficient differentially private algorithm that
accurately answers all the queries. They also proved that if $Q$ and $X$ are both
exponentially large, then under a plausible assumption, no efficient
algorithm exists.

We show that, under the same assumption, if either the number of queries or the data universe is of exponential size, then there is no differentially private algorithm that answers all the queries. Specifically, we prove that if one-way functions and indistinguishability obfuscation exist, then:

1) For every $n$, there is a family $Q$ of $\tilde{O}(n^7)$ queries on a data universe $X$ of size $2^d$ such that no $poly(n,d)$ time differentially private algorithm takes a dataset $D \in X^n$ and outputs accurate answers to every query in $Q$.

2) For every $n$, there is a family $Q$ of $2^d$ queries on a data universe $X$ of size $\tilde{O}(n^7)$ such that no $poly(n,d)$ time differentially private algorithm takes a dataset $D \in X^n$ and outputs accurate answers to every query in $Q$.

In both cases, the result is nearly quantitatively tight, since there is an efficient differentially private algorithm that answers $\tilde{\Omega}(n^2)$ queries on an exponential size data universe, and one that answers exponentially many queries on a data universe of size $\tilde{\Omega}(n^2)$.

Our proofs build on the connection between hardness results in differential privacy and traitor-tracing schemes (Dwork et al., STOC'09; Ullman, STOC'13). We prove our hardness result for a polynomial size query set (resp., data universe) by showing that they follow from the existence of a special type of traitor-tracing scheme with very short ciphertexts (resp., secret keys), but very weak security guarantees, and then constructing such a scheme.

**Category / Keywords: **differential privacy, traitor-tracing, indistinguishability obfuscation

**Date: **received 20 Jul 2016, last revised 30 May 2018

**Contact author: **luke at cs columbia edu

**Available format(s): **PDF | BibTeX Citation

**Note: **Changed constants in differential privacy requirement to match that of theorem.

**Version: **20180531:005811 (All versions of this report)

**Short URL: **ia.cr/2016/721

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