# Differential privacy and Shamir's scheme

I have been unable to find any proofs (for my own reference) that demonstrate that Shamir's secret sharing scheme does (or does not) satisfy the definition of $$\textbf{differential privacy}$$ as provided below:

$$\mathbb{D}$$ denotes the set of possible databases. $$\textbf{Q}$$ denotes the set of possible queries on the databases. Let $$\textbf{M}$$ denote a mechanism such that: $$$$\label{mechanism_definition} \textbf{M}: \mathbb{D} \times \textbf{Q} \mapsto \mathbb{R}^p$$$$

A mechanism $$\textbf{M}$$ as defined above is said to satisfy differential privacy if: $$$$\label{differential_privacy_equation} \frac{\mathbb{P}[\textbf{M}(D,q) \in S]}{\mathbb{P}[\textbf{M}(D',q) \in S]} = 1$$$$

$$\forall$$ D $$\in$$ $$\mathbb{D}$$ and $$\forall$$ q $$\in$$ $$\textbf{Q}$$.

$$\forall$$ S $$\subset$$ Range ($$\textbf{M}$$).

This is the reference I used for the definition provided.

I would appreciate some guidance in this regard.

• Can you cast shamir secret sharing into this framework as an exercise? – kodlu Jan 21 at 22:27
• The definition you have given for differential privacy is with privacy parameter $\varepsilon=0$. No interesting algorithms satisfy this definition. – Thomas Jan 21 at 23:25