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In pure differential privacy, the parameter $\epsilon$ represents the desired privacy loss. The smaller the $\epsilon$ is, the more privacy we can obtain. What happens when we want the privacy loss $\epsilon = 0$. Does that mean we need to keep the data permanently private and we do not answer any queries from outside ?

Thanks

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Yes.

$\varepsilon$-differential privacy with $\varepsilon=0$ implies that the output of the mechanism must be independent of the input -- i.e., it provides no information whatsoever.

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When we look at the definition of Differencial Privacy in the Dwork papers, e.g. The Algorithmic Foundations of Differential Privacy, we have that a $\epsilon$-Differential Privacy for algorithm $M$ as:

$$Pr[M(x_1) \in S] \leq e^\epsilon Pr[M(x_2) \in S]$$

with $S \subseteq range(M)$, for any two neighboring data sets $x_1,x_2$, differing by the addition or removal of a single row.

If your concern is about an $\epsilon = 0$, and so $e^\epsilon=1$, which means that the probability distribution of $M$ is not influenced by the addition or removal of any single row; in other words, no single row can impact the output of $M$.

Maybe we have several examples, but let us imagine that the size of the data sets $x_1, x_2$ are big enough to make $\epsilon$ negligible, such that the influence of a single row is almost null in the output of $M$.

So, I think that the answer is no: whatever the mechanism $M$ answers, it will leak no information about any individual row.

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