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\begin_body

\begin_layout Title
Differential Privacy Mechanism for 
\family typewriter
Prio3
\family default
 and 
\family typewriter
PureDpDiscreteLaplace
\end_layout

\begin_layout Author
David Cook,
 
\begin_inset CommandInset href
LatexCommand href
name "dcook@divviup.org"
target "dcook@divviup.org"
type "mailto:"
literal "false"

\end_inset


\end_layout

\begin_layout Date
July 18,
 2024
\end_layout

\begin_layout Standard
Recall the definitions of pure differential privacy and the discrete Laplace distribution from 
\begin_inset CommandInset citation
LatexCommand cite
key "CKS20"
literal "false"

\end_inset

,
 and the definition of global sensitivity from 
\begin_inset CommandInset citation
LatexCommand cite
key "NRS07"
literal "false"

\end_inset

.
\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:pure-dp"

\end_inset

A randomized algorithm 
\begin_inset Formula $M:\mathcal{X}^{n}\rightarrow\mathcal{Y}$
\end_inset

 satisfies 
\begin_inset Formula $\varepsilon$
\end_inset

-differential privacy if,
 for all neighboring datasets 
\begin_inset Formula $x,x^{\prime}\in\mathcal{X}^{n}$
\end_inset

 (differing on a single element),
 and all events 
\begin_inset Formula $E\subseteq\mathcal{Y}$
\end_inset

,
 we have 
\begin_inset Formula $\mathbb{P}\left[M\left(x\right)\in E\right]\leq e^{\varepsilon}\cdot\mathbb{P}\left[M\left(x^{\prime}\right)\in E\right]$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:discrete-laplace"

\end_inset

The discrete Laplace distribution,
 with scale parameter 
\begin_inset Formula $t$
\end_inset

,
 is defined by the following probability density function,
 supported on the integers.
\begin_inset Formula 
\[
\forall x\in\mathbb{Z},\underset{X\leftarrow\mathrm{Lap}_{\mathbb{Z}}\left(t\right)}{\mathbb{P}}\left[X=x\right]=\frac{e^{\nicefrac{1}{t}}-1}{e^{\nicefrac{1}{t}}+1}\cdot e^{\nicefrac{-\left|x\right|}{t}}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Separator plain
\end_inset


\end_layout

\begin_layout Definition
\begin_inset CommandInset label
LatexCommand label
name "def:global-sensitivity"

\end_inset

The global sensitivity of a query function 
\begin_inset Formula $f\left(x\right)$
\end_inset

 on a dataset 
\begin_inset Formula $x$
\end_inset

 is the maximum distance between two query outputs over any neighboring datasets.
 Here,
 we will use the 
\begin_inset Formula $\ell_{1}$
\end_inset

 metric to measure distances between results,
 and the replacement definition of neighboring datasets.
\begin_inset Formula 
\[
GS_{f}=\underset{x,x^{\prime}:\mathrm{neighboring}}{\mathrm{max}}\left\Vert f\left(x\right)-f\left(x^{\prime}\right)\right\Vert _{1}
\]

\end_inset


\end_layout

\begin_layout Standard
The following differential privacy mechanism is implemented for the combination of the 
\family typewriter
PureDpDiscreteLaplace
\family default
 strategy and the 
\family typewriter
Prio3Histogram
\family default
 or 
\family typewriter
Prio3SumVec
\family default
 VDAFs.
 Let 
\begin_inset Formula $f\left(x\right)$
\end_inset

 be the VDAF's aggregation function,
 operating over the integers.
 The aggregation function produces a query result 
\begin_inset Formula $q=f\left(x\right)\in\mathcal{Y}$
\end_inset

.
 Without loss of generality,
 we assume the domain 
\begin_inset Formula $\mathcal{Y}$
\end_inset

 is a vector of integers,
 
\begin_inset Formula $\mathcal{Y}=\mathbb{Z}^{d}$
\end_inset

.
 Let 
\begin_inset Formula $\mathbb{F}_{p}$
\end_inset

 be field of prime order over which 
\family typewriter
Prio3
\family default
 operates.
 Noise is sampled from the discrete Laplace distribution 
\begin_inset Formula $\mathrm{Lap}_{\mathbb{Z}}\left(\nicefrac{GS_{f}}{\varepsilon}\right)$
\end_inset

,
 projected into the field,
 and added to each coordinate of aggregate share field element vectors.
 Let 
\begin_inset Formula $\pi_{\mathbb{F}_{p}}:\mathbb{Z}\rightarrow\mathbb{F}_{p}$
\end_inset

 and 
\begin_inset Formula $\pi_{\mathbb{Z}}:\mathbb{F}_{p}\rightarrow\mathbb{Z}$
\end_inset

 be the natural projections between the integers and field elements,
 where 
\family roman
\series medium
\shape up
\size normal
\emph off
\nospellcheck off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
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\color none

\begin_inset Formula $\pi_{\mathbb{Z}}$
\end_inset


\family default
\series default
\shape default
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\xout default
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\color inherit
 maps field elements to 
\begin_inset Formula $\left[0,p\right)$
\end_inset

.
 Let 
\begin_inset Formula $\vec{\pi}_{\mathbb{F}_{p}}:\mathbb{Z}^{d}\rightarrow\mathbb{F}_{p}^{d}$
\end_inset

 be the natural extension to project vectors of integers into vectors of field elements.
 Let 
\begin_inset Formula $\vec{q}^{*}=f^{*}\left(x\right)\in\mathbb{F}_{p}^{d}$
\end_inset

 be the element-wise projections of 
\begin_inset Formula $\vec{q}$
\end_inset

 and 
\begin_inset Formula $f$
\end_inset

 into the field using 
\begin_inset Formula $\vec{\pi}_{\mathbb{F}_{p}}$
\end_inset

.
 The un-noised aggregate shares produced by 
\family typewriter
Prio3
\family default
 are secret shares of the query result,
 
\begin_inset Formula $\vec{q}^{*}=\vec{q}^{\left(0\right)}+\vec{q}^{\left(1\right)}$
\end_inset

.
 Each aggregator samples noise from the discrete Laplace distribution and adds it to the un-noised aggregate shares,
 and then sends the sum as their aggregate share to the collector.
 If we pessimistically assume that only one honest aggregator out of the two aggregators is adding differential privacy noise,
 then the mechanism produces 
\begin_inset Formula $\vec{M}\left(x\right)=\vec{q}^{\left(0\right)}+\vec{q}^{\left(1\right)}+\vec{\pi}_{\mathbb{F}_{p}}\left(\vec{Z}\right)=\vec{q}^{*}+\vec{\pi}_{\mathbb{F}_{p}}\left(\vec{Z}\right)$
\end_inset

,
 where 
\begin_inset Formula $Z_{j}\leftarrow\mathrm{Lap}_{\mathbb{Z}}\left(\nicefrac{GS_{f}}{\varepsilon}\right)$
\end_inset

 is drawn independently for all 
\begin_inset Formula $1\leq j\le d$
\end_inset

.
\end_layout

\begin_layout Theorem
\begin_inset CommandInset label
LatexCommand label
name "thm:multivariate-laplace-mechanism-satisfies-pure-dp"

\end_inset


\begin_inset Formula $\vec{M}\left(x\right)=\vec{\pi}_{\mathbb{F}_{p}}\left(f\left(x\right)\right)+\vec{\pi}_{\mathbb{F}_{p}}\left(\vec{Z}\right),Z_{j}\leftarrow Lap_{\mathbb{Z}}\left(\nicefrac{GS_{f}}{\varepsilon}\right)$
\end_inset

 satisfies 
\begin_inset Formula $\varepsilon$
\end_inset

-differential privacy.
\end_layout

\begin_layout Proof
We will show Definition 
\begin_inset CommandInset ref
LatexCommand formatted
reference "def:pure-dp"
plural "false"
caps "false"
noprefix "false"
nolink "false"

\end_inset

 holds for singleton events,
 where 
\begin_inset Formula $E$
\end_inset

 is a set of cardinality one,
 then other events will follow by a union bound.
\end_layout

\begin_layout Proof
For neighboring datasets 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $x^{\prime}$
\end_inset

,
 let 
\begin_inset Formula $\vec{q}=f\left(x\right)$
\end_inset

,
 
\begin_inset Formula $\vec{q}^{\prime}=f\left(x^{\prime}\right)$
\end_inset

,
 and 
\begin_inset Formula $\vec{q}^{*}=\vec{\pi}_{\mathbb{F}_{p}}\left(f\left(x\right)\right)$
\end_inset

,
 and let 
\begin_inset Formula $q_{j}$
\end_inset

,
 
\begin_inset Formula $q_{j}^{*}$
\end_inset

,
 and 
\begin_inset Formula $Z_{j}$
\end_inset

 denote the 
\begin_inset Formula $j$
\end_inset

-th component of the respective vectors.
 Then 
\begin_inset Formula $M_{j}\left(x\right)=q_{j}^{*}+\pi_{\mathbb{F}_{p}}\left(Z_{j}\right)$
\end_inset

.
 Applying the probability density function of the discrete Laplace distribution,
 we have:
\begin_inset Formula 
\[
\forall j\in\left[d\right],y_{j}\in\mathbb{F}_{p},\mathbb{P}\left[M_{j}\left(x\right)=y_{j}\right]=\mathbb{P}\left[\pi_{\mathbb{F}_{p}}\left(Z_{j}\right)=y_{j}-q_{j}^{*}\right]
\]

\end_inset


\begin_inset Formula 
\[
=\stackrel[k=-\infty]{\infty}{\sum}\mathbb{P}\left[Z_{j}=\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right]
\]

\end_inset


\begin_inset Formula 
\[
=\stackrel[k=-\infty]{\infty}{\sum}\frac{e^{\nicefrac{\varepsilon}{GS_{f}}}-1}{e^{\nicefrac{\varepsilon}{GS_{f}}}+1}\mathrm{exp}\left(\frac{-\varepsilon\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|}{GS_{f}}\right)
\]

\end_inset


\begin_inset Formula 
\[
=\frac{e^{\nicefrac{\varepsilon}{GS_{f}}}-1}{e^{\nicefrac{\varepsilon}{GS_{f}}}+1}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(\frac{-\varepsilon\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|}{GS_{f}}\right)
\]

\end_inset


\end_layout

\begin_layout Proof
Since each 
\begin_inset Formula $Z_{j}$
\end_inset

 is drawn independently,
 the probability of the mechanism returning some result can be found by taking the product of the probabilities for each dimension of the result vector.
\begin_inset Formula 
\[
\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}\right]=\left(\frac{e^{\nicefrac{\varepsilon}{GS_{f}}}-1}{e^{\nicefrac{\varepsilon}{GS_{f}}}+1}\right)^{d}\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(\frac{-\varepsilon\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|}{GS_{f}}\right)
\]

\end_inset


\begin_inset Formula 
\[
\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}\right]=\left(\frac{e^{\nicefrac{\varepsilon}{GS_{f}}}-1}{e^{\nicefrac{\varepsilon}{GS_{f}}}+1}\right)^{d}\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(\frac{-\varepsilon\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|}{GS_{f}}\right)
\]

\end_inset


\end_layout

\begin_layout Proof
By the definition of global sensitivity,
 we know 
\begin_inset Formula $\left\Vert \vec{q}-\vec{q}^{\prime}\right\Vert _{\ell_{1}}\le GS_{f}$
\end_inset

.
 We can break up the 
\begin_inset Formula $\ell_{1}$
\end_inset

 distance between 
\begin_inset Formula $\vec{q}$
\end_inset

 and 
\begin_inset Formula $\vec{q}^{\prime}$
\end_inset

 by dimension,
 and relate this sum of absolute values of differences to the product of multiplicative factors of 
\begin_inset Formula $e^{\left|q_{j}-q_{j}^{\prime}\right|}$
\end_inset

,
 in order to get the bound we need.
 Let 
\begin_inset Formula $\delta_{j}=q_{j}-q_{j}^{\prime}$
\end_inset

.
 By the triangle inequality,
 
\begin_inset Formula $\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\le\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|+\left|\delta_{j}\right|$
\end_inset

.
 Since 
\begin_inset Formula $\varepsilon>0$
\end_inset

 and 
\begin_inset Formula $GS_{f}>0$
\end_inset

,
 then,
\begin_inset Formula 
\[
-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\ge-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|-\frac{\varepsilon}{GS_{f}}\left|\delta_{j}\right|
\]

\end_inset


\begin_inset Formula 
\[
\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)\ge\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|-\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}\right)
\]

\end_inset


\begin_inset Formula 
\[
\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)\ge\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)e^{-\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}}
\]

\end_inset


\begin_inset Formula 
\[
e^{\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)\ge\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)
\]

\end_inset


\begin_inset Formula 
\[
\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le e^{\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\]

\end_inset

Since the above holds for a fixed 
\begin_inset Formula $y$
\end_inset

,
 
\begin_inset Formula $q$
\end_inset

 and 
\begin_inset Formula $q^{\prime}$
\end_inset

,
 and any 
\begin_inset Formula $j$
\end_inset

 and 
\begin_inset Formula $k$
\end_inset

,
 we can first add and then multiply inequalities together.
\begin_inset Formula 
\begin{multline*}
\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
e^{\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
\stackrel[j=1]{d}{\prod}e^{\frac{\varepsilon\left|\delta_{j}\right|}{GS_{f}}}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
\mathrm{exp}\left(\frac{\varepsilon\stackrel[j=1]{d}{\sum}\left|\delta_{j}\right|}{GS_{f}}\right)\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
\mathrm{exp}\left(\frac{\varepsilon}{GS_{f}}\left\Vert \vec{q}-\vec{q}^{\prime}\right\Vert _{\ell_{1}}\right)\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset

Then,
 since 
\begin_inset Formula $\left\Vert \vec{q}-\vec{q}^{\prime}\right\Vert _{\ell_{1}}\le GS_{f}$
\end_inset

,
\begin_inset Formula 
\begin{multline*}
\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
e^{\frac{\varepsilon}{GS_{f}}GS_{f}}\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset


\begin_inset Formula 
\begin{multline*}
\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}+kp\right|\right)\le\\
e^{\varepsilon}\stackrel[j=1]{d}{\prod}\stackrel[k=-\infty]{\infty}{\sum}\mathrm{exp}\left(-\frac{\varepsilon}{GS_{f}}\left|\pi_{\mathbb{Z}}\left(y_{j}\right)-q_{j}^{\prime}+kp\right|\right)
\end{multline*}

\end_inset

This shows that,
 for any neighboring 
\begin_inset Formula $x$
\end_inset

 and 
\begin_inset Formula $x^{\prime}$
\end_inset

,
 and any 
\begin_inset Formula $y$
\end_inset

,
 
\begin_inset Formula $\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}\right]\le e^{\varepsilon}\cdot\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}\right]$
\end_inset

.
\end_layout

\begin_layout Proof
Next,
 we apply union bounds.
 For any event 
\begin_inset Formula $E\subseteq\mathcal{Y}$
\end_inset

,
 we decompose the probabilities into that of the corresponding singleton events.
 (note that the singleton events are mutually exclusive)
\begin_inset Formula 
\[
\mathbb{P}\left[\vec{M}\left(x\right)\in E\right]=\mathbb{P}\left[\vec{M}\left(x\right)\in\underset{\vec{y}_{i}\in E}{\bigcup}\vec{y}_{i}\right]=\underset{\vec{y}_{i}\in E}{\sum}\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}_{i}\right]
\]

\end_inset


\begin_inset Formula 
\[
\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)\in E\right]=\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)\in\underset{\vec{y}_{i}\in E}{\bigcup}\vec{y}_{i}\right]=\underset{\vec{y}_{i}\in E}{\sum}\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}_{i}\right]
\]

\end_inset

Since we already know 
\begin_inset Formula $\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}\right]\le e^{\varepsilon}\cdot\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}\right]$
\end_inset

 for all 
\begin_inset Formula $\vec{y}$
\end_inset

,
 we can add multiple such inequalities together.
\begin_inset Formula 
\[
\underset{\vec{y}_{i}\in E}{\sum}\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}_{i}\right]\le\underset{\vec{y}_{i}\in E}{\sum}e^{\varepsilon}\cdot\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}_{i}\right]
\]

\end_inset


\begin_inset Formula 
\[
\underset{\vec{y}_{i}\in E}{\sum}\mathbb{P}\left[\vec{M}\left(x\right)=\vec{y}_{i}\right]\le e^{\varepsilon}\underset{\vec{y}_{i}\in E}{\sum}\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)=\vec{y}_{i}\right]
\]

\end_inset


\begin_inset Formula 
\[
\mathbb{P}\left[\vec{M}\left(x\right)\in E\right]\le e^{\varepsilon}\cdot\mathbb{P}\left[\vec{M}\left(x^{\prime}\right)\in E\right]
\]

\end_inset

Therefore,
 
\begin_inset Formula $\vec{M}\left(x\right)$
\end_inset

 satisfies 
\begin_inset Formula $\varepsilon$
\end_inset

-differential privacy.
\end_layout

\begin_layout Standard
We will now apply this mechanism to 
\family typewriter
Prio3Histogram
\family default
 and 
\family typewriter
Prio3SumVec
\family default
 in turn.
\end_layout

\begin_layout Standard
First,
 let the 
\family typewriter
length
\family default
 parameter of 
\family typewriter
Prio3Histogram
\family default
 be denoted by 
\begin_inset Formula $l$
\end_inset

.
 Then,
 each measurement making up a dataset is an element of 
\begin_inset Formula $\mathcal{X}=\left\{ 0,1,2,\dots,l-1\right\} $
\end_inset

,
 and the query result is a vector of counts,
 in 
\begin_inset Formula $\mathcal{Y}=\mathbb{Z}_{\ge0}^{l}$
\end_inset

.
 The VDAF's aggregation function is our query function,
 
\begin_inset Formula $f\left(x\right)$
\end_inset

.
 It maps each measurement to a one-hot vector,
 with the position of the one determined by the measurement,
 and adds them up.
 The global sensitivity of this query function is 
\begin_inset Formula $GS_{f}=2$
\end_inset

.
 When one measurement in a dataset is replaced with another,
 then either the result is unchanged,
 or one count is decreased by one and another is increased by one.
 Thus,
 the 
\family typewriter
scale
\family default
 parameter is 
\begin_inset Formula $t=\nicefrac{2}{\varepsilon}$
\end_inset

,
 and the mechanism will add noise drawn independently from 
\begin_inset Formula $\mathrm{Lap}_{\mathbb{Z}}\left(\nicefrac{2}{\varepsilon}\right)$
\end_inset

 to each counter in both aggregate shares.
\end_layout

\begin_layout Standard
Let the 
\family typewriter
length
\family default
 parameter of 
\family typewriter
Prio3SumVec
\family default
 be denoted by 
\begin_inset Formula $l$
\end_inset

,
 and the 
\family typewriter
bits
\family default
 parameter be denoted by 
\begin_inset Formula $b$
\end_inset

.
 Each measurement making up a dataset is an element of 
\begin_inset Formula $\mathcal{X}=\left\{ 0,1,2,\dots,2^{b}-1\right\} ^{l}$
\end_inset

.
 The query result is a vector of sums,
 in 
\begin_inset Formula $\mathcal{Y}=\mathbb{Z}_{\ge0}^{l}$
\end_inset

.
 The VDAF's aggregation function is our query function,
 
\begin_inset Formula $f\left(x\right)=\stackrel[i=1]{n}{\sum}x_{i}$
\end_inset

.
 The global sensitivity of this query function is 
\begin_inset Formula $GS_{f}=\left(2^{b}-1\right)\cdot l$
\end_inset

,
 because substituting one measurement may increase or decrease each component of the vector sum by up to 
\begin_inset Formula $2^{b}-1$
\end_inset

.
 Thus,
 the 
\family typewriter
scale
\family default
 parameter is 
\begin_inset Formula $t=\frac{\left(2^{b}-1\right)l}{\varepsilon}$
\end_inset

,
 and the mechanism will add noise drawn independently from 
\begin_inset Formula $\mathrm{Lap}_{\mathbb{Z}}\left(\frac{\left(2^{b}-1\right)l}{\varepsilon}\right)$
\end_inset

 to each sum in both aggregate shares.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "CKS20"
literal "false"

\end_inset

Canonne,
 C.
 L.,
 Kamath,
 G.,
 and Steinke,
 T.,
 
\begin_inset Quotes eld
\end_inset

The Discrete Gaussian for Differential Privacy
\begin_inset Quotes erd
\end_inset

,
 2020,
 <
\begin_inset CommandInset href
LatexCommand href
target "https://arxiv.org/abs/2004.00010"

\end_inset

>.
\end_layout

\begin_layout Bibliography
\begin_inset CommandInset bibitem
LatexCommand bibitem
key "NRS07"
literal "false"

\end_inset

Nissim,
 K.,
 Raskhodnikova,
 S.,
 and Smith,
 A.,
 
\begin_inset Quotes eld
\end_inset

Smooth sensitivity and sampling in private data analysis
\begin_inset Quotes erd
\end_inset

,
 2007,
 <
\begin_inset CommandInset href
LatexCommand href
target "https://cs-people.bu.edu/ads22/pubs/NRS07/NRS07-full-draft-v1.pdf"
literal "false"

\end_inset

>.
\end_layout

\end_body
\end_document
