Can a folded LaPlace distribution (or other folded distributions) be used with Ɛ-differential privacy

Question

I have a single value in (or over) our dataset, let's say a count of something, and we want to keep that value private within a certain range. This range is the sensitivity.

The adversary can ask if a certain value it provides is lower or higher than the value in our dataset, but there is one catch. We cannot answer that the value is lower when it is higher than the actual value.

My idea is to use a folded LaPlace distribution (or any other folded distribution) to sample a value that is by definition lower than or equal to the actual value, and then answer lower or higher to the adversary, based on the sampled value.

I am uncertain whether Ɛ-differential privacy still holds with such a distribution.

In a broader sense I am interested in Ɛ-differential privacy where the noise added is bounded by logical constraints, but I must be using the wrong search terms because I can't find a paper on this.

Answer 1

To answer my own question: No, this cannot be done. It is a form of pre-processing and that breaks differential privacy.

The (obvious) intuition here is that if you only add noise in one direction, you still release an awful lot of information, namely that the real value cannot be situated on one side of the released value: Consider a situation where you only deduct noise from the real value. When releasing that value, the real value cannot be less than that value, because that would mean that you would have added noise.