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thething: Added pf:thething-prv

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Manos Katsomallos
2021-09-19 23:02:00 +02:00
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text/problem/thething/problem.tex
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@ -170,7 +170,33 @@ Theorem~\ref{theor:thething-prv} proposes how to achieve the desired privacy for
\end{theorem}
% \mk{To discuss.}
Due to space constraints, we omit the proof of Theorem~\ref{theor:thething-prv} and defer it for a longer version of this paper.
% Due to space constraints, we omit the proof of Theorem~\ref{theor:thething-prv} and defer it for a longer version of this paper.
\begin{proof}
\label{pf:thething-prv}
All mechanisms use independent randomness, and therefore for a series of events $S_T = {D_1, \dots, D_T}$ and outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$ it holds that
$$Pr[\mathcal{M}(S_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D_i) = \pmb{o}_i]$$
Likewise, for any {\thething}-neighboring series of events $S'_T$ of $S_T$ with the same outputs $(\pmb{o}_1, \dots, \pmb{o}_T) \in O \subseteq \mathcal{O}$
$$Pr[\mathcal{M}(S'_T) = (\pmb{o}_1, \dots, \pmb{o}_T)] = \prod_{i \in [1, T]} Pr[\mathcal{M}_i(D'_i) = \pmb{o}_i]$$
Since $S_T$ and $S'_T$ are {\thething}-neighboring, there exists $i \in T$ such that $D_i = D'_i$ for a set of {\thethings} with timestamps $L$.
Thus, we get
$$\frac{Pr[\mathcal{M}(S_T) = (\pmb{o}_1, \dots, \pmb{o}_T)]}{Pr[\mathcal{M}(S'_T) = (\pmb{o}_1, \dots, \pmb{o}_T)]} = \prod_{i \in L \cup \{t\}} \frac{Pr[\mathcal{M}_i(D_i) = \pmb{o}_i]}{Pr[\mathcal{M}_i(D'_i) = \pmb{o}_i]}$$
$D_i$ and $D'_i$ are neighboring for $i \in L \cup \{t\}$.
$\mathcal{M}_i$ is differential private and from Definition~\ref{def:dp} we get that $\frac{Pr[\mathcal{M}_i(D_i) = \pmb{o}_i]}{Pr[\mathcal{M}_i(D'_i) = \pmb{o}_i]} \leq e^{\varepsilon_i}$.
Hence, we can write
$$\frac{Pr[\mathcal{M}(S_T) = (\pmb{o}_1, \dots, \pmb{o}_T)]}{Pr[\mathcal{M}(S'_T) = (\pmb{o}_1, \dots, \pmb{o}_T)]} \leq \prod_{i \in L \cup \{t\}} e^{\varepsilon_i} = e^{\sum_{i \in L \cup \{t\}} \varepsilon_i}$$
For any $O \in \mathcal{O}$ we get $\frac{Pr[\mathcal{M}(S_T) \in O}{Pr[\mathcal{M}(S'_T) \in O]} \leq e^{\sum_{i \in L \cup \{t\}} \varepsilon_i}$.
If the formula of Theorem~\ref{theor:thething-prv} holds, then we get $\frac{Pr[\mathcal{M}(S_T) \in O}{Pr[\mathcal{M}(S'_T) \in O]} \leq e^\varepsilon$.
Due to Definition~\ref{def:thething-prv} this concludes our proof.
\end{proof}
\subsubsection{{\Thething} privacy mechanisms}
\label{subsec:lmdk-mechs}