# ($$\epsilon$$,$$\delta$$)-Differential Privacy#

## In brief#

A relaxed version of Differential Privacy, named ($$\epsilon$$,$$\delta$$)-Differential Privacy, allows a little privacy loss ($$\delta$$) due to a variation in the output distribution for the privacy mechanism.

## More in Detail#

### Relaxing $$\epsilon$$-Differential Privacy#

The ($$\epsilon$$,$$\delta$$)-differential privacy model is a common relaxation of $$\epsilon$$-differential privacy. Under the $$\epsilon$$-differential privacy model, the probabilities that the function $$\mathtt{f}$$ outputs the same output when computed over neighboring datasets are allowed to diverge up to an $$e^\epsilon$$ factor. The ($$\epsilon$$,$$\delta$$)-differential privacy model additionally tolerates the two probabilities to diverge by a small additional quantity, denoted $$\delta$$.

This leads to revisiting the formal definition of $$\epsilon$$-differential privacy as follows. A random function $$\mathtt{f}$$ with range $$\mathcal{O}$$ satisfies ($$\epsilon$$, $$\delta$$)-differential privacy if and only if for all possible pairs of datasets ($$\mathcal{D}$$, $$\mathcal{D}'$$) such that $$\mathcal{D}'$$ is $$\mathcal{D}$$ with one record more or one record less, and for all $$\mathcal{S} \subseteq \mathcal{O}$$, then it holds that: $$\mathtt{Pr} [ \mathtt{f} ( \mathcal{D} ) \in \mathcal{S} ] \leq e^\mathbf{\epsilon} \times \mathtt{Pr} [ \mathtt{f} ( \mathcal{D}' ) \in \mathcal{S} ] + \delta$$ where $$\epsilon>0$$ and $$\delta \geq 0$$ are the privacy parameters. When $$\delta>0$$, ($$\epsilon$$,$$\delta$$)-differential privacy is also called approximate differential privacy.

## Bibliography#

The ($$\epsilon$$,$$\delta$$)-differential privacy model is introduced in  and thoroughly studied in .

1

Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni Naor. Our data, ourselves: privacy via distributed noise generation. In EUROCRYPT. 2006.

2

Sebastian Meiser. Approximate and probabilistic differential privacy definitions. IACR Cryptol. ePrint Arch., 2018:277, 2018.

This entry was written by Tristan Allard.