# Can Differential Privacy be used to show that two distributions are indistinguishable?

Differential privacy can be used to show that the "privacy loss" of a certain computation is "bounded" in a meaningful way. In cryptography, often "indistinguishability" is considered, i.e. we want $$\Delta(D_0, D_1)$$ to be small. Can these two concepts be related?

I know vaguely that standard notions of differential privacy can be related to things like "concentrated differential privacy", which itself is a Renyi-type bound on two distributions (which is somewhat similar to the total variation distance bound that I want). But I want to end my proof with "So, the two distributions are indistinguishable" --- can differential privacy help me with this goal?

• Can you specify the "renyi type" bound you're talking about. A quick look at Dwork's paper shows bounds depending on subgaussian properties. Mar 16 '21 at 1:08
• @kodlu Later papers by Bun and Steinke recast things in terms of Renyi-divergences I think. I'm mostly going on information from this paper.
– Mark
Mar 16 '21 at 1:33

These sorts of arguments come up in the theory of randomness extractors where we argue that a random function from a certain family when applied to inputs with a certain minimum entropy ($$H_\infty$$, the -ve log-maximum probability of an outcome in the Renyi notation) produce outputs that are indistinguishable from uniform up to some $$\epsilon$$ bound. The differential privacy of the uniform and extractor distributions can be bounded in terms of input size, output size and $$H_\infty$$ (see for example the Kamp and Zuckerman theorem of the article).

The min-entropy should be the choice of entropy function when using differential privacy as it bounds the worst case scenario of concentration of inputs.