text:
\section{Introduction}
In the Private Information Retrieval (PIR) problem, a database of $K$ messages are replicated at $N$ servers. There is a user who wishes to retrieve a single or multiple messages belonging to the database while protecting the identity of the demanded message(s) from any individual server~\cite{chor1995private,sun2017capacity,banawan2018capacity,sun2017}. In order to retrieve the desired message(s), the user generates one query for each server. Upon receiving the user's query, each server will return an answer to the user, which depends on the stored messages and the received query. To ensure that each server learns nothing about the identity of the message(s) being retrieved by the user, in an information theoretic sense, each query must be marginally independent of the desired message(s) index.

In a single-server setting or a multi-server setting when all servers can fully collude, the user must download the whole database to achieve privacy in the information-theoretic sense~\cite{chor1995private}. However, when the user has some side information about the messages in the database \cite{kadhe2017private,heidarzadeh,li2018single,chen2017capacity,li2018converse,shariatpanahi2018multi,heidarzadeh2018capacity,heidarzadeh2019capacity,kazemi2019multi,tandon2017capacity,wei2018cache,wei2018fundamental,heidarzadeh2019single,kazemi2019single} or when the servers do not fully collude~{\cite{sun2017capacity,banawan2018capacity,sun2017}}, the privacy can be achieved in a more efficient manner in terms of minimizing the download cost (i.e., the amount of information downloaded
from the server(s)).

For the PIR problem in the presence of side information, two different types of privacy can be considered: (i)~\emph{$W$-privacy}, which requires that the identity of the user's demanded message(s) be protected, and (ii)~\emph{$(W,S)$-privacy}, which requires that  the identities of both the user's demanded message(s) and the message(s) in the user's side information be protected. When the side information is a random subset of messages, the problem is referred to as \emph{PIR with Side Information (PIR-SI)} or \emph{PIR with Private Side Information (PIR-PSI)} where $W$-privacy or $(W,S)$-privacy is required, respectively. The single-server settings of these problems were studied in~\cite{kadhe2017private,heidarzadeh,li2018single}, and their multi-server settings were studied in~\cite{li2018converse,chen2017capacity,shariatpanahi2018multi}. In~\cite{heidarzadeh2018capacity} and \cite{heidarzadeh2019capacity}, we studied the single-server setting of a related problem in which the side information is a random linear combination of a random subset of messages. This problem is referred to as \textit{PIR with Coded Side Information (PIR-CSI)} or \textit{PIR with Private Coded Side Information (PIR-PCSI)} when $W$-privacy or $(W,S)$-privacy is required, respectively. Also,  in \cite{kazemi2019multi}, we recently studied the multi-server setting of the PIR-CSI problem.

In this work, we consider the multi-server setting of the PIR-PCSI problem. In this setting, a database of $K$ messages is replicated across $N$ servers, and a user, who knows a random linear combination of a random subset of $M$ messages in the database, wishes to obtain a message by sending queries to the servers. The goal is to design a scheme that protects the identities of both the user's demanded message and the messages forming the user's side information, while minimizes the download cost. The servers are assumed to know the number of messages contributing to the user's side information beforehand. However, the indices and the coefficients of the messages in the user's side information are not known to the servers in advance. The motivation for this type of side information comes from several practical scenarios. For instance, the side information could have been obtained in advance from a trusted server with limited knowledge about the database, or through overhearing in a wireless network, or from the information locally stored in the user's cache.

\subsection{Main Contributions}

We consider two settings of the PIR-PCSI problem depending on whether the user's demanded message is one of the messages forming the user's side information or not. We characterize (or derive a lower bound on) the capacity of each setting, where the capacity is defined as the supremum of all achievable rates (i.e., the inverse of the normalized download cost).
In the first setting, the message demanded by the user is not one of the messages forming the user's side information. For this setting, we prove that the capacity is equal to $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. Interestingly, the capacity in this setting is equal to the capacity of multi-server PIR-PSI problem~\cite{chen2017capacity} in which $M$ uncoded messages are available at the user as side information. This result shows that there is no loss in capacity due to restricting the user's side information to one random linear combination of $M$ messages, instead of $M$ uncoded messages.

The converse proof readily follows from the fact that the capacity of this setting is upper-bounded by the capacity of the multi-server PIR-PSI which is given by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$ (see~\cite[Theorem~1]{chen2017capacity}).

For the achievability proof, we devise a new protocol that builds upon two existing achievability schemes for two different problems: (i) the Private Computation (PC) scheme of~\cite{sun2018capacity} for multi-server private computation where a user wishes to privately retrieve one arbitrary linear combination of the messages replicated at multiple servers, and (ii) our Specialized GRS Code scheme proposed in~\cite{heidarzadeh2019capacity} for single-server PIR-PCSI.

The main ideas of our achievability scheme are as follows. First, the user utilizes the Specialized GRS Code scheme of~\cite{heidarzadeh2019capacity} for single-server PIR-PCSI to construct $K-M$ independent super-messages which are some linearly independent combinations of the original messages, to play the role of the original messages in a multi-server private computation problem. Then, the user and the $N$ servers leverage the PC scheme of~\cite{sun2018capacity} for the constructed $K-M$ super-messages in such a way that the user can privately download one of $\binom{K}{M+1}$ linear combinations of the $K-M$ super-messages where the support of each linear combination is a distinct subset of $[K]$ of size $M+1$.

Additionally, for the setting wherein the demanded message is one of the messages forming the user's side information, we show that the capacity is lower-bounded by ${\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}}$. The proof is based on a new achievability scheme that leverages the PC scheme of~\cite{sun2018capacity} for multi-server private computation, combined with our Modified Specialized GRS Code scheme proposed in~\cite{heidarzadeh2019capacity} for single-server PIR-PCSI.

\section{Problem Formulation}\label{sec:SN}
We denote random variables by bold letters and their realizations by non-bold letters.
 For a positive integer $i$, let $[i]\triangleq \{1,\dots,i\}$. Let $\mathbb{F}_q$ be a finite field for some prime $q$, and let $\mathbb{F}_q^{\times} \triangleq \mathbb{F}_q\setminus \{0\}$ be the multiplicative group of $\mathbb{F}_q$. Let $\mathbb{F}_{q^m}$ be an extension field of $\mathbb{F}_q$ for some integer $m\geq 1$.

Consider $N$ non-colluding identical servers, each of which stores $K$ messages $X_1,\dots,X_K$, where $\mathbf{X}_i$ is independently and uniformly distributed over $\mathbb{F}_{q^m}$, i.e., for all $i\in [K]$, it holds that \[{H(\mathbf{X}_i) = L\triangleq m\log_2 q}, \quad \text{and}\quad  {H(\mathbf{X}_1,\dots,\mathbf{X}_K) = KL}.\]

Suppose that there is a user that wishes to retrieve a message $X_W$ from the servers for some $W\in [K]$, and has a linear combination ${Y^{[S,C]}\triangleq \sum_{i\in S} c_i X_i}$ for some $S \triangleq \{i_1,\dots,i_M\}\in \mathcal{S}$ and some ${C \triangleq \{c_{i_1},\dots,c_{i_M}\} \in \mathcal{C}}$, where $\mathcal{S}$ is the set of all $M$-subsets of $[K]$, and $\mathcal{C}$ is the set of all length-$M$ sequences with elements from $\mathbb{F}^{\times}_q$. We call $W$ the \emph{demand index}, $X_W$ the \emph{demand}, $Y^{[S,C]}$ the \emph{side information}, $S$ the \emph{side information index set}, and $M$ the \emph{side information size}.

We assume that $\mathbf{S}$ is uniformly distributed over $\mathcal{S}$, and that $\mathbf{C}$ is uniformly distributed over $\mathcal{C}$. Also, two different models for the conditional distribution of $\mathbf{W}$ given $\mathbf{S}=S$ are considered:
\begin{itemize}
    \item {Model I}: $\mathbf{W}$ is uniformly distributed over $[K]\setminus S$;
    \item {Model II}: $\mathbf{W}$ is uniformly distributed over $S$.
\end{itemize}
\begin{comment}
\begin{equation*}
p_{\mathbf{W}|\mathbf{S}}(W|S) =
\left\{\begin{array}{ll}
\frac{1}{K-M}, & W\not\in S,\\
0, & \text{otherwise}.
\end{array}\right.
\end{equation*}
\end{comment}
\begin{comment}
\begin{equation*}
p_{\mathbf{W}|\mathbf{S}}(W|S) =
\left\{\begin{array}{ll}
\frac{1}{M}, & W\in S,\\
0, & \text{otherwise};
\end{array}\right.
\end{equation*}
\end{comment}

It is assumed that $1\leq M \leq K-1$ and $2\leq M\leq K$ for Model I and Model II, respectively. Note that for both models it holds that $\mathbf{W}$ is uniformly distributed over $[K]$.
We assume that no server knows the realizations of $\mathbf{S},\mathbf{C},\mathbf{W}$ in advance. In contrast, we assume that all servers know the considered model (i.e., whether $\mathbf{W}\not\in \mathbf{S}$ or $\mathbf{W}\in \mathbf{S}$), the side information size $M$,  the distributions of $\mathbf{S}$ and $\mathbf{C}$, and the conditional distribution of $\mathbf{W}$ given $\mathbf{S}$.

For any $S$, $C$, $W$, in order to retrieve $X_W$, the user generates $N$ queries $Q_1^{[W,S,C]},\dots, Q_N^{[W,S,C]}$, and sends to the $n$-th server the query $Q_n^{[W,S,C]}$. Each query $Q_n^{[W,S,C]}$ is assumed to be a (potentially stochastic) function of $W$, $S$, $C$, and $Y^{[S,C]}$. Upon receiving the query $Q_n^{[W,S,C]}$, the $n$-th server responds to the user with an answer $A_n^{[W,S,C]}$. The answer $A_n^{[W,S,C]}$ is a (deterministic) function of the query $Q_n^{[W,S,C]}$ and the messages in $X_{[K]}\triangleq \{X_1,\dots,X_K\}$. Note that for all $n\in [N]$, it holds that

\[(\mathbf{W},\mathbf{S}) \rightarrow (\mathbf{Q}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]},\mathbf{X}_{[K]}) \rightarrow \mathbf{A}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]}\] forms a Markov chain, and \[H(\mathbf{A}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]}| \mathbf{Q}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]},\mathbf{X}_{[K]}) = 0.\]

The answers $A_1^{[W,S,C]},\dots,A_N^{[W,S,C]}$ from all servers along with the side information $Y^{[S,C]}$ and the queries $Q_1^{[W,S,C]},\dots,Q_N^{[W,S,C]}$ must enable the user to retrieve the demand $X_W$, i.e.,
\[H(\mathbf{X}_{\mathbf{W}}| \mathbf{A}^{[\mathbf{W},\mathbf{S},\mathbf{C}]},\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}, \mathbf{Y}^{[\mathbf{S},\mathbf{C}]},\mathbf{W},\mathbf{S},\mathbf{C})=0,\]
where $A^{[W,S,C]}\triangleq \{A_1^{[W,S,C]},\dots,A_N^{[W,S,C]}\}$, and $Q^{[W,S,C]}\triangleq \{Q_1^{[W,S,C]},\dots,Q_N^{[W,S,C]}\}$. This condition is referred to as the \emph{recoverability condition}.

In addition, the queries $Q_1^{[W,S,C]}, \dots,Q_N^{[W,S,C]}$ must not reveal any information about the user's demand index $W$ and side information index set $S$ to any server,
\[I(\mathbf{W},\mathbf{S}; \mathbf{Q}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]},\mathbf{A}_n^{[\mathbf{W},\mathbf{S},\mathbf{C}]},\mathbf{X}_{[K]})=0\quad \forall n \in [N].\] This condition is referred to as the \emph{$(W,S)$-privacy condition.}

For both models (Model I and Model II), we would like to design a protocol for generating queries $\{Q_1^{[W,S,C]},\dots,Q_N^{[W,S,C]}\}$ for any given $W,S,C$. The protocol also prescribes, for all $n \in [N]$, how the $n$-th server generates the answer $A_n^{[W,S,C]}$, given $Q_n^{[W,S,C]}$ and $X_{[K]}$.

A protocol that satisfies both the $(W,S)$-privacy and recoverability conditions for all $W,S,C$ with $W\not\in S$ (or $W\in S$), is referred to as a \emph{PIR-PCSI--I} (or \emph{PIR-PCSI--II}) \emph{protocol}. The problem of designing a PIR-PCSI--I (or PIR-PCSI--II) protocol is referred to as the \emph{PIR-PCSI--I} (or \emph{PIR-PCSI--II}) \emph{problem}.

The \emph{rate} of a {PIR-PCSI--I} or {PIR-PCSI--II} protocol is defined as the ratio of the entropy of a message, i.e., $L$, to the total entropy of answers from all servers, i.e., $H(\mathbf{A}^{[\mathbf{W},\mathbf{S},\mathbf{C}]})$.

The \emph{capacity} of the PIR-PCSI--I (PIR-PCSI--II) problem is defined as the supremum of rates over all PIR-PCSI--I (PIR-PCSI--II) protocols. We denote by ${C_{(W,S)-\text{\it I}}}$ the capacity of the PIR-PCSI--I problem, and denote by ${C_{(W,S)-\text{\it II}}}$ the capacity of the PIR-PCSI--II problem.

In this work, our goal is to characterize (or derive lower bounds on) ${C_{(W,S)-\text{\it I}}}$ and ${C_{(W,S)-\text{\it II}}}$, and to design {PIR-PCSI--I} and {PIR-PCSI--II} protocols that achieve the capacity (or the derived lower bound on the capacity).

\section{Main Results}

In this section, we present our main results. Theorem~\ref{thm:PIRPCSI-I} characterizes the capacity of the PIR-PCSI--I problem ${C_{(W,S)-\text{\it I}}}$, and Theorem~\ref{thm:PIRPCSI-II} presents a lower-bound on the capacity of the PIR-PCSI--II problem ${C_{(W,S)-\text{\it II}}}$. The proofs of theorems~\ref{thm:PIRPCSI-I} and~\ref{thm:PIRPCSI-II} are given in sections~\ref{sec:PIR-PCSI-I} and~\ref{sec:PIR-PCSI-II}, respectively.

\begin{theorem}\label{thm:PIRPCSI-I}
The capacity of the PIR-PCSI--I problem with $N$ servers, $K$ messages, and side information size ${1\leq M \leq K-1}$ is given by
\[
{C_{(W,S)-\text{\it I}}}= \left(1+\frac{1}{N}+\dots+\frac{1}{N^{K-M-1}}\right)^{-1}.\]

\end{theorem}

Interestingly, this result indicates that the capacity of multi-server PIR-PCSI--I, i.e., ${C_{(W,S)-\text{\it I}}}$, is equal to the capacity of the multi-server PIR-PSI~\cite{chen2017capacity} where $M$ uncoded messages are available at the user as side information. Note that having only a random linear combination of $M$ messages as side information instead of $M$ uncoded messages, cannot increase the capacity which implies the converse. Thus, to complete the proof of Theorem~\ref{thm:PIRPCSI-I}, we only need to prove the achievability which is presented in Section~\ref{sec:PIR-PCSI-I}.
Notably, our results show that having only one random linear combination of messages instead of multiple uncoded messages does not decrease the capacity, either.

\begin{theorem}\label{thm:PIRPCSI-II}
The capacity of the PIR-PCSI--II problem with $N$ servers, $K$ messages, and side information size ${2 \leq M\leq K}$ is lower-bounded by
\[
{C_{(W,S)-\text{\it II}}} \geq \left(1+\frac{1}{N}+\dots+\frac{1}{N^{K-M}}\right)^{-1}.
\]

\end{theorem}

This result is interesting because it shows that the lower-bound on the capacity of the multi-server PIR-PCSI--II is the same as the capacity of multi-server PIR-SI when the size of side information is $M-1$. That is,
having a side information which is only a random linear combination of $M$ messages including the demanded message would be at least as effective as knowing $M-1$ messages separately in terms of minimizing the download cost.
For the proof, we construct a {PIR-PCSI--II} protocol that achieves the capacity lower-bound of Theorem~\ref{thm:PIRPCSI-II}. It should be noted that the tightness of this lower bound remains open in general.

\section{The ~PIR-PCSI-I ~Problem}\label{sec:PIR-PCSI-I}

In this section, we complete the proof of Theorem~\ref{thm:PIRPCSI-I} by proposing an achievability scheme for arbitrary $N$, ${K\geq 1}$ and ${0\leq M\leq K-1}$ that achieves the rate $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. The proposed protocol, referred to as the \emph{Multi-Server PIR-PCSI--I protocol}, is a non-trivial combination of the Specialized GRS Code scheme of~\cite{heidarzadeh2019capacity} for single-server PIR-PCSI problem and the Private Computation (PC) scheme of~\cite{sun2018capacity} for multi-server private computation problem.

For the proposed protocol, we assume that $q\geq K$, and each message $X_i$ consists of $m= N^{\binom{K}{M+1}}$ symbols over $\mathbb{F}_q$.\vspace{0.2cm}

\textbf{Multi-Server PIR-PCSI--I protocol:} The protocol consists of the following five steps:

\textit{\textbf{Step 1:}} The user utilizes the Specialized GRS Code protocol proposed in \cite{heidarzadeh2019capacity} to first construct a polynomial ${p(x) = \sum_{i=0}^{K-M-1} p_i x^i \triangleq \prod_{i\not\in S\cup W} (x-\omega_i)}$ where $\omega_1,\dots,\omega_K$ are $K$ arbitrarily chosen distinct elements from $\mathbb{F}_q$, and then construct $r \triangleq K-M$ vectors $\underline{u}_1,\dots,\underline{u}_{r}$, each of length $K$, such that ${\underline{u}_{i} = [\beta_{1}\omega_1^{i-1} ,\dots,\beta_{K}\omega_K^{i-1}]}$ for $i\in [r]$, where $\beta_j=\frac{c_j}{p(\omega_j)}$ for $j\in S$, and $\beta_j$ is a randomly chosen element from $\mathbb{F}_q^{\times}$ for $j\not\in S$.

\textit{\textbf{Step 2:}} Let $\hat{X}_i\triangleq \sum_{j=1}^{K} \beta_j\omega_j^{i-1} X_{j}$ for $i\in [r]$. Each $\hat{X}_i$ is referred to as a \emph{super-message}. Note that the vector $\underline{u}_i$ (constructed in Step~1) is the vector of coefficients of the messages $\{X_i\}_{i\in [K]}$ in the super-message $\hat{X}_i$. Let $F\triangleq\binom{K}{M+1}$, and let $J_1,J_2,\dots,J_F$ be the collection of all $(M+1)$-subsets of $[K]$ in a lexicographical order. The structure of the Specialized GRS Code protocol \cite{heidarzadeh2019capacity} ensures that for each $J_f$, $f \in [F]$, there exist exactly $q-1$ linear combinations $Z^1_f,Z^2_f,\dots,Z^{q-1}_f$ of the messages $\{X_i\}_{i \in J_f}$ with (non-zero) coefficients from $\mathbb{F}^{\times}_q$, such that for every $k\in [q-1]$, $Z^k_f$ can be written as a linear combination of the super-messages $\hat{X}_1,\dots,\hat{X}_r$. Let $\underline{v}^{k}_f\triangleq [v^{k}_{f,1},\dots,v^{k}_{f,r}]$ be a vector of length $r$ such that $Z^{k}_f = \sum_{i=1}^{r} v^{k}_{f,i} \hat{X}_i$. Note that, for each $f \in [F]$, $Z^1_f,Z^2_f,\dots,Z^{q-1}_f$ are the same up to a scalar multiple, i.e., for each $k\in [q-1]$, $Z^{k}_f = \alpha_k Z^{1}_f$, or equivalently, $\underline{v}^{k}_f = \alpha_k \underline{v}^{1}_f$, for some distinct $\alpha_k\in \mathbb{F}^{\times}_q$.
For each $f\in [F]$, let $i_f\triangleq \min(J_f)$. Note also that for every $f\in [F]$, there exists a unique $k_f\in [q-1]$ such that the coefficient of the message $X_{i_f}$ in the linear combination $Z^{k_f}_{f}$ is equal to $1$. The user then constructs $F$ vectors $\underline{v}_1,\dots,\underline{v}_F$, each of length $r$, such that $\underline{v}_f = \underline{v}^{k_f}_f$. (Note that the above procedure dictates a specific choice of the coefficient vectors $\underline{v}_f$. However, for each $f\in [F]$, the vector $\underline{v}_f$ can be chosen arbitrarily from the set of vectors $\{\underline{v}^{k}_f\}_{k\in [q-1]}$.) Let $Z_f\triangleq Z^{k_f}_f$ for $f\in [F]$. Each $Z_f$ is referred to as a (linear) \emph{function}. Note that $\underline{v}_f$ is the vector of coefficients of the super-messages $\{\hat{X}_i\}_{i\in [r]}$ in the function $Z_f$.

\textit{\textbf{Step 3:}} The user sends to all servers the vectors $\underline{u}_1,\dots,\underline{u}_{r}$ (associated with the super-messages $\hat{X}_1,\dots,\hat{X}_r$), and the vectors $\underline{v}_1,\dots,\underline{v}_{F}$ (associated with the functions $Z_1,\dots,Z_F$). It is noteworthy that the user needs only to send the vectors $\{\underline{u}_i\}_{i\in [r]}$ to all servers, and each server can construct the vectors $\{\underline{v}_f\}_{f\in [F]}$ by using $\{\underline{u}_i\}_{i\in [r]}$ (according to the procedure described in Step~2).

\textit{\textbf{Step 4:}} The user and the servers leverage the PC scheme of~\cite{sun2018capacity} with $r$ (independent) messages and $F$ (linear) functions of these messages in order for the user to privately retrieve one of these functions. In particular, the $r=K-M$ super-messages $\{\hat{X}_i\}_{i\in [r]}$ and the $F$ functions $\{Z_f\}_{f\in [F]}$ play the role of the original messages and the functions in the PC scheme, respectively, and the user is interested in retrieving the function $Z_{f^{*}}$ privately, where $Z_{f^{*}}$ is an $\mathbb{F}^{\times}_q$-linear combination (i.e., a linear combination with non-zero coefficients only) of the messages $\{X_i\}_{i \in W\cup S}$. (By the construction, there exists one (and only one) function $Z_f$ among $Z_1,\dots,Z_F$ such that $Z_f$ is an $\mathbb{F}_q^{\times}$-linear combination of the messages $\{X_i\}_{i \in W\cup S}$.) To be more specific, each server first constructs the super-messages $\{\hat{X}_i\}_{i\in [r]}$ by using the coefficient vectors $\{\underline{u}_i\}_{i\in [r]}$ (defined in Step~3), and then constructs the functions $\{Z_f\}_{f\in [F]}$ by using the super-messages $\{\hat{X}_i\}_{i\in [r]}$ and the coefficient vectors $\{\underline{v}_f\}_{f\in [F]}$ (defined in Step~3). Note that each function $Z_f$ for $f\in [F]$ consists of $m=N^F$ $\mathbb{F}_q$-symbols where $N$ is the number of servers. Then, each server sends to the user $m({{1}/{N}+{1}/{N^2}+\dots+{1}/{N^{K-M}}})$ carefully designed linear combinations of all $\mathbb{F}_q$-symbols associated with all functions $\{Z_f\}_{f\in [F]}$. The details of the design of the user's query to each server as well as the linear combinations transmitted by each server (which also depend on the query of the user) can be found in~\cite[Section~4]{sun2018capacity}.

\begin{comment}
The user and the $N$ servers leverage the PC protocol for $r$ independent super-messages in such a way that the user can download one of $F=\binom{K}{M+1}$ linear combinations $\{\widetilde{W}_f\}_{f\in [F]}$ of the $r$ independent super-messages, \alex{defined as} $ {\widetilde{W}_f=\sum_{k=1}^{r} \alpha_{f,k} \hat{X}_k}$ for $f \in [F]$, where the support of each linear combination $\widetilde{W}_f$ for $f \in [F]$ is a distinct subset of $[K]$ of size $M+1$. \alex{How the values of  $\alpha_{f,k}$ are defined?} It should be noted that the construction by the Specialized GRS Code Protocol \cite{heidarzadeh2019capacity} ensures that there exists a set of coefficients $\alpha_{f,k}$ for all $f \in [F]$ and $k \in [r]$ such that the set of the supports of the functions $\widetilde{W}_f$ for all $f \in [F]$ created based on these coefficients, be equal to the set of all subsets of $[K]$ of size $M+1$. Specifically, the construction by the Specialized GRS Code Protocol guarantees that there exists a specific set of coefficients $\{\alpha_{f,1},\alpha_{f,2},\dots,\alpha_{f,r}\}$ for some $f \in [F]$, say $f=g$, where $\alpha_{g,k}=0$ for $k \not \in (S\cup W)$, $\alpha_{g,k}=c_k$ for $k \in S$, and $\alpha_{g,k}\neq 0$ for $k=W$. \alexn{I do not understand this sentence - it requires more clarifications.} The user retrieves $X_W$ by subtracting off the contribution of its side information $Y^{[S,C]}$ from $\widetilde{W}_g=\sum_{k=1}^{r} \alpha_{g,k} \hat{X}_k$.\vspace{0.2cm}
\end{comment}

\begin{comment}
\begin{example}\label{exam1}(PC Protocol \cite{sun2018capacity}) Suppose there are $N = 2$ servers and $K = 2$ messages. Suppose the user wishes to privately compute one of the following $F = 4$ functions.\vspace{0.2cm}

$\begin{array}{lcl}
\widetilde{W}_1=2X_1+X_2,~~~ \widetilde{W}_2=X_1+2X_2\\
\widetilde{W}_3=X_1+4X_2,~~~ \widetilde{W}_4=X_2\\
\end{array}$\vspace{0.2cm}

Suppose the user is interested in retrieving $\widetilde{W}_1$. The protocol assumes that each of the messages consists of ${L = N^F= 16}$ bits. Let $\pi$ represents a permutation over ${[L]}$. Let $u_f(i)=\sigma_i \widetilde{W}_f(\pi(i))$ for all $f \in [F]$, ${i \in [L]}$, where $\widetilde{W}_f(\pi(i))$ are the bits from message $\widetilde{W}_f$, permuted by $\pi$, and $u_f(i)$ are the corresponding signed versions obtained by scaling with $\sigma_i \in \{+1,-1\}$. For this example, let $(a_i,b_i,c_i,d_i)=(u_1(i),u_2(i),u_3(i),u_4(i))$ for all ${i \in [L]}$. It should be noted that unlike the Sun-Jafar Protocol presented in \cite{sun2017capacity} where independent random permutations are applied to the bits of each message, in PC Protocol, the same permutation $\pi$ is applied to the bits of all messages. Also, the same sign variable $\sigma_i$ is applied to the bits from different messages that have the same index. Both $\pi$ and $\sigma_i$ are generated privately, independently and uniformly by the user such that they are not known to the
servers. The user will start with the query structure of the Sun-Jafar Protocol, and then modifies the query according to the \textbf{Index assignment} and \textbf{Sign assignment} processes, which are both elaborated in \cite{sun2018capacity}. Then, it requests the bits from the first server ($S_1$) and the second server ($S_2$) as given in Table \ref{table1}.\vspace{-0.05cm}

\begin{table}[ht]
    \centering
    \scalebox{1.2}{
\begin{tabular}{ |c|c| }
 \hline
 $S_1$ & $S_2$ \\ [0.7ex]
 \hline
 $a_{1},b_{1},c_{1},d_{1}$ & $a_{2},b_{2},c_{2},d_{2}$ \\ [0.7ex]
 \hline
 $a_{3}-b_{2}$ & $a_{6}-b_{1}$\\ [0.7ex]
 $a_{4}-c_{2}$ & $a_{7}-c_{1}$\\ [0.7ex]
 $a_{5}-d_{2}$ & $a_{8}-d_{1}$ \\ [0.7ex]
 \hline
  $b_{4}-c_{3}$ & $b_{7}-c_{6}$\\ [0.7ex]
  $b_{5}-d_{3}$ & $b_{8}-d_{6}$\\ [0.7ex]
  $c_{5}-d_{4}$ & $c_{8}-d_{7}$\\ [0.7ex]
 \hline
  $a_{9}-b_{7}+c_{6}$ & $a_{12}-b_{4}+c_{3}$\\ [0.7ex]
  $a_{10}-b_{8}+d_{6}$ & $a_{13}-b_{5}+d_{3}$\\ [0.7ex]
  $a_{11}-c_{8}+d_{7}$ & $a_{14}-c_{5}+d_{4}$\\ [0.7ex]
  \hline
  $b_{11}-c_{10}+d_{9}$ & $b_{14}-c_{13}+d_{12}$\\ [0.7ex]
  \hline
  $a_{15}-b_{14}+c_{13}-d_{12}$ & $a_{16}-b_{11}+c_{10}-d_{9}$\\ [0.7ex]
  \hline
\end{tabular}}
\caption{The query table for $N=2, K= 2, F = 4$ \cite{sun2018capacity}.}
\label{table:1}
    \label{table1}
\end{table} \vspace{-0.6cm}

As can be seen, $15$ symbols are queried from each server. However, based on the information obtained from the other server, $3$ symbols are redundant. Thus, the user can download $12$ generic combinations of these $15$ symbols. The PC Protocol satisfies both the recoverability and $W$-Privacy conditions. Because, the user can decode the $16$ bits of $W_1$ by downloading $24$ bits ($12$ bits per server), and each of the two servers cannot distinguish which function is requested by the user.
\end{example}
\end{comment}

\textbf{\textbf{Example 1.}} (Multi-Server PIR-PCSI--I protocol) Assume that there are $N = 2$ servers, $K=4$ messages from $\mathbb{F}_{5^{16}}$, and $M=2$. Note that each message consists of ${m= N^{\binom{K}{M+1}}= 16}$ symbols from $\mathbb{F}_{5}$. Suppose that the user demands the message $X_1$ and has a coded side information $Y={X_2+X_3}$, i.e., $W = 1$, $S= \{2,3\}$, and ${C = \{1,1\}}$ (i.e., ${c_{2}=1, c_{3}=1}$).

First, the user picks ${K=4}$ distinct elements $\omega_1,\dots,\omega_4$ from $\mathbb{F}_5$. Suppose that the user chooses $\omega_1=0$, $\omega_2=1$, $\omega_3=2$, $\omega_4=3$. Then, the user constructs the polynomial ${{p(x) = \prod_{i\not\in S\cup W} (x-\omega_i)}=x-\omega_4=x-3}$. The user then computes $\beta_j$ for $j\in S$, i.e., $\beta_2$ and $\beta_3$, by setting ${\beta_2=\frac{c_2}{p(\omega_2)}=2}$ and ${\beta_3=\frac{c_3}{p(\omega_3)}=4}$, and chooses $\beta_j$ for $j\not\in S$, i.e., $\beta_1$ and $\beta_4$, at random (from $\mathbb{F}^{\times}_5$). Assume that the user chooses $\beta_1=1$ and $\beta_4=2$. Then, the user constructs ${r=K-M=2}$ vectors $\underline{u}_1$ and $\underline{u}_2$, each of length ${K=4}$, such that ${\underline{u}_i=[\beta_1\omega_1^{i-1},\dots,\beta_K\omega_K^{i-1}]}$ for $i\in \{1,2\}$. That is, the user constructs ${\underline{u}_1=[1,2,4,2]}$ and ${\underline{u}_2=[0,2,3,1]}$. For set $J_1=\{1,2,3\}$, there exist exactly ${q-1=4}$ vectors ${\underline{v}^{k}_1 = [k,3k]}$ for ${k\in \{1,\dots,4\}}$ such that ${k\underline{u}_1+3k\underline{u}_2=k[1,3,3,0]}$.

It should be noted that there exists no other vector ${\underline{v}=[v_1,v_2]}$ such that the support of the vector ${v_1\underline{u}_1 + v_2\underline{u}_2}$ is ${J_1 = \{1,2,3\}}$. Note that the coefficient of the message ${X_{i_1}=X_1}$ (i.e., ${i_1=\min(J_1)=1}$) in the \mbox{function} ${Z_1}$ is equal to $1$ when ${k=1}$. Thus, the user constructs the vector ${\underline{v}_1=\underline{v}^{1}_1=[1,3]}$. Similarly, the user \mbox{constructs} the vectors ${\underline{v}_2=[1,2]}$, ${\underline{v}_3=[1,4]}$ and ${\underline{v}_4=[0,3]}$. Then, the user sends to all servers the vectors $\underline{u}_1$ and $\underline{u}_{2}$ (associated with the super-messages $\hat{X}_1$ and $\hat{X}_2$), and the vectors ${\underline{v}_1,\dots,\underline{v}_{4}}$ (associated with the functions ${Z_1,\dots,Z_4}$). Using the coefficient vectors $\underline{u}_1$ and $\underline{u}_{2}$, each server first constructs the two super-messages ${\hat{X}_1=X_1+2X_2+4X_3+2X_4}$ and ${\hat{X}_2=2X_2+3X_3+X_4}$. Then, it constructs the functions ${Z_1,\dots,Z_4}$ using the super-messages $\hat{X}_1$ and $\hat{X}_2$ and the coefficient vectors ${\underline{v}_1,\dots,\underline{v}_{4}}$ as follows:
\[
\begin{array}{lcl}
Z_1=\hat{X}_1+3\hat{X}_2=X_1+3X_2+3X_3\\
Z_2=\hat{X}_1+2\hat{X}_2=X_1+X_2+4X_4\\
Z_3=\hat{X}_1+4\hat{X}_2=X_1+X_3+X_4\\
Z_4=3\hat{X}_2=X_2+4X_3+3X_4\\
\end{array}
\]

Finally, the user and the servers apply the PC scheme of~\cite{sun2018capacity} for two super-messages $\hat{X}_1$, $\hat{X}_2$ in order for the user to privately retrieve the function ${Z_1}$. (Note that among the functions $Z_1,\dots,Z_4$, only $Z_1$ is an $\mathbb{F}^{\times}_5$-linear combination of the messages $\{X_i\}_{i\in W\cup S} = \{X_1,X_2,X_3\}$.) The details of the PC scheme for this example are as follows. Let $\pi: [16]\rightarrow [16]$ be a randomly chosen permutation. Let $u_f(i)\triangleq\sigma_i Z_f(\pi(i))$ for $f \in [4]$ and ${i \in [16]}$, where $Z_f(\pi(i))$ is the $\pi(i)$-th $\mathbb{F}_5$-symbol of $Z_f$, and $\sigma_i$ is a randomly chosen element from $\{-1,+1\}$.
For simplifying the notation, let $(a_i,b_i,c_i,d_i)=(u_1(i),u_2(i),u_3(i),u_4(i))$ for all ${i \in [16]}$. The user then queries $15$ carefully designed linear combinations of the symbols $\{\{a_i\}_{i\in [16]},\{b_i\}_{i\in [16]},\{c_i\}_{i\in [16]},\{d_i\}_{i\in [16]}\}$, as given in Table~\ref{table1} \cite{sun2018capacity}, from each of the servers (S1 and S2).

As shown in~\cite{sun2018capacity}, among the $15$ symbols queried from S1 (or S2), based on the information obtained from S2 (or S1), $3$ symbols are redundant. For instance, consider the $15$ symbols queried from S1. (Similar observations can be made regarding the queries from S2.) Among the $4$ symbols $\{a_1,b_1,c_1,d_1\}$, any $2$ symbols suffice to recover the other $2$ symbols. For example, $c_1$ and $d_1$ can be obtained from $a_1$ and $b_1$. (Note that $Z_3$ and $Z_4$ can be written as a linear combination of $Z_1$ and $Z_2$.) Thus, the server S1 needs to send two arbitrary symbols from $\{a_1,b_1,c_1,d_1\}$. In addition, given any $2$ symbols from $\{a_2,b_2,c_2,d_2\}$, any $5$ symbols among the $6$ symbols $\{{a_3-b_2},{a_4-c_2},{a_5-d_2},{b_4-c_3},{b_5-d_3},{c_5-d_4}\}$ queried from S1 would suffice to recover the remaining symbol. For example, ${c_5-d_4}$ can be obtained from the symbols $\{a_3-b_2,{a_4-c_2},{a_5-d_2},{b_4-c_3},{b_5-d_3},b_2,d_2\}$ (for details, see~\cite[Section~5.1]{sun2018capacity}).
Thus, each of the servers S1 and S2 needs to send to the user only $12$ symbols. In particular, S1 transmits $2$ arbitrary symbols from $\{a_1,b_1,c_1,d_1\}$, $5$ arbitrary symbols from $\{{a_3-b_2},{a_4-c_2},{a_5-d_2},{b_4-c_3},{b_5-d_3},{c_5-d_4}\}$, and the $4$ symbols $\{a_9-b_7+c_6,a_{10}-b_8+d_6,a_{11}-c_8+d_7,b_{11}-c_{10}+d_9\}$, and the symbol $\{a_{15}-b_{14}+c_{13}-d_{12}\}$; and S2 transmits $2$ arbitrary symbols from $\{a_2,b_2,c_2,d_2\}$, $5$ arbitrary symbols from $\{a_6-b_1,{a_7-c_1},{a_8-d_1},{b_7-c_6},{b_8-d_6},{c_8-d_7}\}$, and the $4$ symbols $\{a_{12}-b_4+c_3,a_{13}-b_5+d_3,a_{14}-c_5+d_4,b_{14}-c_{13}+d_{12}\}$, and the symbol $\{a_{16}-b_{11}+c_{10}-d_{9}\}$.

\begin{comment}
\begin{table}[ht]
    \centering
    \scalebox{1.01}{
\begin{tabular}{ |c|c| }
 \hline
 $DB_1$ & $DB_2$ \\ [0.8ex]
 \hline
 $a_{1},b_{1},c_{1},d_{1}$ & $a_{2},b_{2},c_{2},d_{2}$ \\ [0.8ex]
 \hline
 $a_{3}-b_{2}$, $a_{4}-c_{2}$, $a_{5}-d_{2}$& $a_{6}-b_{1}$, $a_{7}-c_{1}$, $a_{8}-d_{1}$\\ [0.8ex]
  $b_{4}-c_{3}$, $b_{5}-d_{3}$, $c_{5}-d_{4}$ & $b_{7}-c_{6}$, $b_{8}-d_{6}$, $c_{8}-d_{7}$\\ [0.8ex]
 \hline
  $a_{9}-b_{7}+c_{6}$, $a_{10}-b_{8}+d_{6}$ & $a_{12}-b_{4}+c_{3}$, $a_{13}-b_{5}+d_{3}$\\ [0.8ex]
  $a_{11}-c_{8}+d_{7}$, $b_{11}-c_{10}+d_{9}$ & $a_{14}-c_{5}+d_{4}$, $b_{14}-c_{13}+d_{12}$\\ [0.8ex]
  \hline
  $a_{15}-b_{14}+c_{13}-d_{12}$ & $a_{16}-b_{11}+c_{10}-d_{9}$\\ [0.8ex]
  \hline
\end{tabular}}
\caption{The queries of the PC protocol for $N=2$, $K=2$ super-messages, $F=4$, when the user demands ${\widetilde{W}_1}$\cite{sun2018capacity}.}
\label{table:1}
    \label{table1}
\end{table}
\end{comment}

\begin{table}[t!]
    \caption{The queries of the PC protocol for $N=2$, $2$ super-messages, $F=4$, when the user demands ${Z_1}$\cite{sun2018capacity}.}
    \label{table1}
    \centering
    \scalebox{1.25}{
\begin{tabular}{ |c|c| }
 \hline
 S1 & S2 \\
 \hline
 $a_{1},b_{1},c_{1},d_{1}$ & $a_{2}, b_{2},c_{2},d_{2}$ \\
 \hline
 $a_{3}-b_{2}$ & $a_{6}-b_{1}$\\
 $a_{4}-c_{2}$ & $a_{7}-c_{1}$\\
 $a_{5}-d_{2}$ & $a_{8}-d_{1}$ \\

  $b_{4}-c_{3}$ & $b_{7}-c_{6}$\\
  $b_{5}-d_{3}$ & $b_{8}-d_{6}$\\
  $c_{5}-d_{4}$ & $c_{8}-d_{7}$\\
 \hline
  $a_{9}-b_{7}+c_{6}$ & $a_{12}-b_{4}+c_{3}$\\
  $a_{10}-b_{8}+d_{6}$ & $a_{13}-b_{5}+d_{3}$\\
  $a_{11}-c_{8}+d_{7}$ & $a_{14}-c_{5}+d_{4}$\\

  $b_{11}-c_{10}+d_{9}$ & $b_{14}-c_{13}+d_{12}$\\
  \hline
  $a_{15}-b_{14}+c_{13}-d_{12}$ & $a_{16}-b_{11}+c_{10}-d_{9}$\\
  \hline
\end{tabular}}
\end{table}

From the answers by the servers, the user obtains all $16$ symbols $a_1,\dots,a_{16}$, and accordingly, all $16$ symbols of $Z_1$. (Note that $a_i = u_1(i)=\sigma_i Z_1(\pi(i))$ for $i\in [16]$.) From $Z_1$ ($=X_1+3X_2+3X_3$), the user can decode the desired message $X_1$ by subtracting off the contribution of their side information $X_2+X_3$.

In order to retrieve $X_1$ which consists of $16$ symbols (over $\mathbb{F}_5$), according to the proposed protocol, the user downloads $24$ symbols (over $\mathbb{F}_5$) from both servers, and hence the rate of the proposed protocol is $16/24=2/3$.

Note that for every $3$-subset $\{X_{j_1},X_{j_2},X_{j_3}\}$ of the messages $\{X_i\}_{i\in [4]}$, in the proposed protocol there exists one (and only one) linear combination $Z_f$ for some $f \in [4]$ of the messages $X_{j_1},X_{j_2},X_{j_3}$. On the other hand, the PC scheme guarantees that no server can obtain any information about the index ($f$) of the linear combination $Z_f$ being requested by the user. Thus, the proposed scheme satisfies the $(W,S)$-privacy condition, as desired.

\begin{lemma}\label{lemma1}
The Multi-Server PIR-PCSI--I protocol satisfies the recoverability and (W,S)-privacy conditions, and achieves the rate ${C_{(W,S)-\text{\it I}}}= \left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$.
\end{lemma}

\begin{proof}

Since the messages $\mathbf{X}_{[K]}$ are uniformly and independently distributed over $\mathbb{F}_{q^m}$, and
$\{\hat{X}_1,\dots,\hat{X}_r\}$ are linearly independent combinations of the messages in $X_{[K]}$, thus $\{\hat{\mathbf{X}}_1,\dots,\hat{\mathbf{X}}_r\}$ are uniformly and independently distributed over $\mathbb{F}_{q^m}$ as well, i.e., ${H(\hat{\mathbf{X}}_1) = \dots = H(\hat{\mathbf{X}}_r) = m\log q=L}$. Hence, the rate of the Multi-Server PIR-PCSI--I protocol is the same as the rate of the PC protocol for $N$ servers and ${K-M}$ messages, which is given by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$ (see \cite[Theorem~1]{sun2018capacity}).

From the step $4$ of the Multi-Server PIR-PCSI--I protocol, it is evident that the recoverability condition is satisfied.
The proof of the $(W,S)$-privacy of the proposed protocol is as follows.
The PC protocol protects the privacy of the function (linear combination) requested by the user. That is, given the query, no server can obtain any information about the index of the function requested by the user. Consider an arbitrary server $n\in [N]$, and an arbitrary query $Q_n$ to server $n$, generated by the proposed protocol. Thus, given ${\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n}$, from the perspective of server $n$, every function ${Z_f}$ for ${f \in [F]}$ is equally likely to include the demanded message. We denote the support of $Z_f$ by $\mathcal{Z}_f$, i.e., $\mathcal{Z}_f$ is the set of all indices $i\in [K]$ such that $X_i$ has a non-zero coefficient in the linear combination $Z_f$. Thus, for all $f\in [F]$, we have
\begin{equation}\label{eq:P1}
\Pr(\mathbf{W}\in \mathcal{Z}_f|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n) = \frac{1}{\binom{K}{M+1}},
\end{equation} noting that $F=\binom{K}{M+1}$.
Note that any given index $W'\in [K]$ is in the support of exactly $\binom{K-1}{M}$ functions $Z_f$, $f\in [F]$.
For any given $f\in [F]$, given $\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n$ and $\mathbf{W}\in \mathcal{Z}_f$, from the perspective of server $n$, every index $W'\in \mathcal{Z}_f$ is equally likely to be the demand index. That is, for all $f\in [F]$, we have
\begin{equation}\label{eq:P2}
\Pr(\mathbf{W}=W'|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n,\mathbf{W}\in \mathcal{Z}_f) = \begin{cases} \frac{1}{M+1}, & W'\in \mathcal{Z}_f,\\ 0, & \text{otherwise}.\end{cases}
\end{equation} Furthermore, for any given $f\in [F]$ and $W'\in \mathcal{Z}_f$, we have
\begin{equation}\label{eq:P3}
\Pr(\mathbf{S}=S'|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n,\mathbf{W}\in \mathcal{Z}_f,\mathbf{W} = W') = \begin{cases} 1, & S'= \mathcal{Z}_f\setminus \{W'\},\\ 0, & \text{otherwise}.\end{cases}
\end{equation} Consider arbitrary $W'\in [K]$ and $S'\subset [K]\setminus \{W'\}, |S'| = M$. Let $f'\in [F]$ be the (unique) index such that $\mathcal{Z}_{f'} = W'\cup S'$. It is easy to see that $\Pr(\mathbf{W}=W',\mathbf{S}=S',\mathbf{W}\in \mathcal{Z}_f|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n) = 0$ for all $f\in [F], f\neq f'$. Thus, by using~\eqref{eq:P1}-\eqref{eq:P3}, we can write
\begin{align}
&\Pr(\mathbf{W}=W',\mathbf{S}=S'|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n)\nonumber \\
& = \sum_{f\in [F]} \Pr(\mathbf{W}=W',\mathbf{S}=S',\mathbf{W}\in \mathcal{Z}_f|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n)\nonumber \\
& = \Pr(\mathbf{W}=W',\mathbf{S}=S',\mathbf{W}\in \mathcal{Z}_{f'}|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n)\nonumber \\
& = \Pr(\mathbf{W}\in \mathcal{Z}_{f'}|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n)\times \Pr(\mathbf{W} = W'|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n, \mathbf{W}\in \mathcal{Z}_{f'})\nonumber\\ & \quad \times \Pr(\mathbf{S} = S'|\mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n = Q_n, \mathbf{W}\in \mathcal{Z}_{f'},\mathbf{W} = W')\nonumber\\
& = \frac{1}{\binom{K}{M+1}}\times \frac{1}{M+1}\times 1 \nonumber \\
& = \frac{M! (K-M-1)!}{K!} \label{eq:P4}
\end{align} On the other hand, we have
\begin{align}
&\Pr(\mathbf{W}= W',\mathbf{S}= S')
\nonumber \\ &= \Pr(\mathbf{W}= W') \times \Pr(\mathbf{S}= S'| \mathbf{W}= W') \nonumber \\ & =\frac{1}{K} \times \frac{1}{\binom{K-1}{M}} \nonumber\\
& = \frac{M! (K-M-1)!}{K!}. \label{eq:P5}
\end{align} From~\eqref{eq:P4} and~\eqref{eq:P5}, for any $W'\in [K]$ and $S'\subset [K]\setminus \{W'\}, |S'|=M$, we have \[\Pr(\mathbf{W}= W',\mathbf{S}= S'| \mathbf{Q}^{[\mathbf{W},\mathbf{S},\mathbf{C}]}_n=Q_n) = \Pr(\mathbf{W}= W',\mathbf{S}= S').\] This completes the proof of $(W,S)$-privacy of the proposed protocol.
\end{proof}

\section{The ~PIR-PCSI-II ~Problem}\label{sec:PIR-PCSI-II}

In this section, we prove the result of Theorem~\ref{thm:PIRPCSI-II}
by constructing a PIR-PCSI--II protocol, referred to as the \emph{Multi-Server PIR-PCSI--II protocol}, for arbitrary $N$, ${K\geq 2}$ and ${2 \leq M\leq K}$ that achieves the rate ${\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}}$.

For the proposed protocol, we assume that $q\geq K$, and each message is comprised of ${m = N^{\binom{K}{M}}}$ symbols over $\mathbb{F}_q$.\vspace{0.2cm}

\textbf{Multi-Server PIR-CSI--II protocol:} The protocol consists of four steps, where the steps 2-4 are the same as the steps 2-4 in the Multi-Server PIR-PCSI--I protocol, except that $M$ is replaced with $M-1$ everywhere. The step~1 of the proposed protocol is as follows:

\textit{\textbf{Step 1:}} The user utilizes the Modified Specialized GRS Code protocol proposed in \cite{heidarzadeh2019capacity} to first construct a polynomial  ${p(x) = \sum_{i=0}^{K-M} p_i x^i \triangleq \prod_{i\not\in S} (x-\omega_i)}$ where $\omega_1,\dots,\omega_K$ are $K$ arbitrarily chosen distinct elements from $\mathbb{F}_q$, and then construct $r \triangleq K-M+1$ vectors ${\underline{u}_1,\dots,\underline{u}_{r}}$, each of length $K$, such that ${\underline{u}_{i} = [\beta_1\omega_1^{i-1} ,\dots,\beta_K\omega_K^{i-1}]}$ for $i\in [r]$, where $\beta_j=\frac{c_j}{p(\omega_j)}$ for $j\in S\setminus W$, $\beta_W=\frac{c}{p(\omega_W)}$ where $c$ is chosen uniformly at random from $\mathbb{F}^{\times}_q\setminus \{c_W\}$, and $\beta_j$ is a randomly chosen element from $\mathbb{F}_q^{\times}$ for $j\not\in S$.

\begin{comment}
\textbf{\textbf{Example 2.}} (Multi-Server PIR-CSI--II protocol) Assume that there are $N = 2$ servers, $K=4$ messages from $\mathbb{F}_{5^{16}}$, and $M=3$. Note that each message consists of ${m= N^{\binom{K}{M}}= 16}$ symbols from a finite field $\mathbb{F}_{5}$. Suppose the user demands the message $X_1$ and has a coded message $X_S={X_1+2X_2+3X_3}$ as side information, i.e., ${W = 1}$, ${S= \{1,2,3\}}$ and ${C = \{1,2,3\}}$ (${c_{1}=1,c_{2}=2, c_{3}=3}$).

In the first step, the user picks ${K=4}$ distinct elements ${\omega_1,\dots,\omega_4}$ from $\mathbb{F}_5$. Let assume the user has chosen ${\omega_1=0,\omega_2=1,\omega_3=2,\omega_4=3}$. Then, the user constructs a polynomial ${{p(x) = \prod_{i\not\in S} (x-\omega_i)}=x-\omega_4=x-3}$. Next, the user computes $\beta_j$ for $j\in S\setminus W$, i.e., $\beta_2$ and $\beta_3$, by setting $\beta_2=\frac{c_2}{p(\omega_2)}=4$ and ${\beta_3=\frac{c_3}{p(\omega_3)}=2}$, and computes $\beta_W$, i.e., $\beta_1$, by choosing $c=2$ uniformly at random from $\mathbb{F}^{\times}_5\setminus \{1\}$ and setting $\beta_1=\frac{c}{p(\omega_W)}=1$. Also, the user chooses $\beta_j$ for $j \not \in S$, i.e., $\beta_4$ at random (from $\mathbb{F}^{\times}_5$). Suppose the user has chosen $\beta_4=3$. Then, the user constructs ${r=K-M+1=2}$ vectors $\mathbf{u}_1$ and $\mathbf{u}_2$, each of length ${K=4}$, such that ${\mathbf{u}_i=[\beta_1\omega_1^{i-1},\dots,\beta_K\omega_K^{i-1}]}$ for $i\in \{1,2\}$. That is, the user constructs ${\mathbf{u}_1=[1,4,2,3]}$ and ${\mathbf{u}_2=[0,4,4,4]}$. According to the similar procedure explained in example 1, the user constructs the vectors ${\mathbf{v}_1=[1,3]}$, ${\mathbf{v}_2=[1,2]}$, ${\mathbf{v}_3=[1,4]}$ and ${\mathbf{v}_4=[0,4]}$. The user sends to all servers the vectors $\mathbf{u}_1,\mathbf{u}_{2}$ (associated with the super-messages $\hat{X}_1,\hat{X}_2$), and the vectors ${\mathbf{v}_1,\dots,\mathbf{v}_{4}}$ (associated with the functions ${Z_1,\dots,Z_4}$). Using the coefficient vectors ${\mathbf{u}_1,\mathbf{u}_{2}}$, each server first constructs $2$ super-messages ${\hat{X}_1=X_1+4X_2+2X_3+3X_4}$ and ${\hat{X}_2=4X_2+4X_3+4X_4}$. Then, each server constructs the functions ${Z_1,\dots,Z_4}$ using the super-messages ${\hat{X}_1,\hat{X}_2}$ and the coefficient vectors ${\mathbf{v}_1,\dots,\mathbf{v}_{4}}$, as follows:  \vspace{0.2cm}

$\begin{array}{lcl}
Z_1=\hat{X}_1+3\hat{X}_2=X_1+X_2+4X_3\\
Z_2=\hat{X}_1+2\hat{X}_2=X_1+2X_2+X_4\\
Z_3=\hat{X}_1+4\hat{X}_2=X_1+3X_3+4X_4\\
Z_4=4\hat{X}_2=X_2+X_3+X_4\\
\end{array}$\vspace{0.2cm}

The last step is that the user and the servers apply the PC scheme of~\cite{sun2018capacity} for two super-messages $\hat{X}_1$, $\hat{X}_2$ in such a way that the user can privately download ${Z_1}$. The form of the queries of the PC scheme is given in Table \ref{table1}. Among the $15$ symbols queried from each server, it can be shown that $3$ symbols are redundant based on the information obtained from the other server. Thus, the user can download only $12$ generic combinations of these $15$ symbols from each server. In order to retrieve $X_1$ which has $16$ symbols, the proposed scheme requires to download $24$ symbols, achieving the rate of $2/3$. Based on the same reasoning as discussed in Example 1, the scheme is $(W,S)$-private.
\end{comment}

\begin{lemma}
The Multi-Server PIR-PCSI--II protocol satisfies the recoverability and (W,S)-privacy conditions, and achieves the rate $\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}$.
\end{lemma}

\begin{proof}
The proof is similar to the proof of Lemma \ref{lemma1}, and hence omitted to avoid repetition.
\end{proof}

\section{Conclusion}
In this paper, we studied the multi-server setting of the Private Information Retrieval with Private Coded Side Information (PIR-PCSI) problem. In this problem, there is a database of $K$ messages replicated across $N$ servers, and there is a user who initially has a random linear combination of a random subset of $M$ messages in the database as side information. The goal of the user is to retrieve one message from the servers, while protecting the \mbox{identities} of both the demand message and the side information messages jointly. We considered two different models for this problem depending on whether the side information is a function of the demand message or not. First, we focused on the setting in which the side information is not a function of the demand message. For this setting, we proved that the capacity is given by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M-1}}\right)^{-1}$. Then, we considered the setting in which the side information is a function of the demand message. For this setting, we show that the capacity is lower-bounded by $\left(1+{1}/{N}+\dots+{1}/{N^{K-M}}\right)^{-1}$. Our proposed achievability schemes are inspired by our recently proposed scheme for the single-server PIR-PCSI problem in conjunction with the scheme proposed by Sun and Jafar for multi-server private computation problem.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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text:
Q: Serializing Object Array Property Values Take the following program:
using System;
using System.Collections.Generic;
using System.Linq;

class Person
{
    public string FirstName { get; set; }
    public string LastName { get; set; }
    public int Age { get; set; }
}

class Program
{
    static void Main()
    {
        var records = new Person[]
        {
            new Person{ FirstName = "John", LastName = "Doe", Age = 32 },
            new Person{ FirstName = "Jane", LastName = "Doe", Age = 27 },
            new Person{ FirstName = "Joe", LastName = "College", Age = 28 }
        };

        Console.WriteLine(string.Join(", ", records.Select(r => new
        {
            FullName = r.FirstName + " " + r.LastName
        })));
    }
}

Its expected output is:
John Doe, Jane Doe, Joe College
But its actual output is:
{ FullName = John Doe }, { FullName = Jane Doe }, { FullName = Joe College }
Is it possible to resolve this from inside the WriteLine?
This is a simplification of a bigger problem I recently encountered, and I need to resolve this from inside the WriteLine because I am performing this "serialization" inside a query; I cannot execute more than one statement.

A: Instead of selecting anonymous type use:
records.Select(r =>  r.FirstName + " " + r.LastName)

So your Console.WriteLine could be:
Console.WriteLine(string.Join(", ", records.Select(r => r.FirstName + " " + r.LastName)));

A: Try this,
Console.WriteLine(string.Join(", ", records.Select(r => String.Format("{0} {1}", r.FirstName, r.LastName) ));

A: You are recreating a dynamic object.  I think you meant something like this:
    Console.WriteLine(string.Join(", ", records.Select(r => r.FirstName + " " + r.LastName)));

A: Just use inline lambda expression as below:
Console.WriteLine(string.Join(", ", records.Select(r => r.FirstName + " " + r.LastName));

A: I would suggest to have a ToString() method. This makes the approach more clean.
public class Person {
    public string FirstName { get; set; }
    public string LastName { get; set; }
    public int Age { get; set; }
    public override string ToString() {
        return string.Format("{0} {1}", FirstName, LastName);
    }
}

and then
Console.WriteLine(string.Join(", ", records.Select(r => r.ToString())));

redpajama_set_name: RedPajamaStackExchange


text:
Norwegian Mapping Authority
Base maps (7)
Norway Digital (3)
Arctic SDI (1)
Large areas of Northern Norway (1)
Large areas of Hordaland county (1)
Parts of the Moere coast (1)
West in the Norwegian Sea (1)
GEBCO (1)
mareano (1)
depth data (1)
terrain models of the seabed (1)
raster models of the seabed (1)
Norwegian Mapping Agency, Hydrographic Service (1)
Norwegian Mapping Authority (9)
Historical archive (2)
Circumpolar Arctic basic map cache
Circumpolar basic map with transparency, combining topographic details from the Norwegian Mapping Authority and the Norwegian Polar Institute south to latitude X (southern France). It also includes map data from the other Arctic countries that is published via the Arctic Spatial Data Infrastructure (Arctic SDI). This map does not include place names and road names – these can be added using the circumpolar place name cache service.
Arctic cascading WMS
This service includes topographical maps on a scale of 1:1 million.
Depth relief polar v02 WMS
A WMS containing shadow reliefs of the ocean depths that also cover northern polar regions. Data from GEBCO_08 Grid, version 20100927, http://www.gebco.net, and the Mareano project, http://www.mareano.no. The grid from Mareano is more up-to-date and also usually closer to Norwegian land areas or in Arctic areas. It is being continually expanded under the auspices of the Mareano project. (http://www.mareano.no/start) * The terrain models are based on modern depth measurements by the use of multi-beam echosounders. * The terrain models are based on the average depth value in the area. The area which this data covers is the seabed along the coast of Northern Norway, north of 67 degrees north, and the parts of the ocean west in the Norwegian Sea. The coverage is limited to the areas which have been measured with modern equipment, that is, multi-beam echosounders. The General Bathymetric Chart of the Oceans (GEBCO) aims to provide the most authoritative, publicly-available bathymetry data sets for the world's oceans. GEBCO operates under the auspices of the International Hydrographic Organization (IHO) and the Intergovernmental Oceanographic Commission (IOC) of UNESCO. The latest version of their grid is from 2014, a global 30 arc-second interval grid. Their one arc-minute interval grid was last updated in 2008. (Source in Norwegian: https://www.gebco.net/data_and_products/gridded_bathymetry_data/) Data from the International Bathymetric Chart of the Arctic Ocean – IBCAO (netsite: www.ibcao.org). Only useful for background information. Reference: Jakobsson, M., Macnab, R., Mayer, M., Anderson, R., Edwards, M., Hatzky, J., Schenke, H-W., and Johnson, P., 2008, An improved bathymetric portrayal of the Arctic Ocean: Implications for ocean modeling and geological, geophysical and oceanographic analyses, v. 35, L07602, Geophysical Research Letters, doi:10.1029/2008GL033520
Arctic administrative units
View service showing administrative units for states in the Arctic SDI.
Polar shaded relief map for depths
Data from the GEBCO_08 Grid, version 20100927, http://www.gebco.net, and the Mareano project, http://www.mareano.no
Circumpolar place names cache
Circumpolar place name map for combining with maps that do not include place names. This is primarily created to be combined with "circumpolar Arctic basic map", but it can also be combined with other maps with no place names or road names.
Arctic SDI cache
The service contains GEBCO, VMAP0 data at higher zoom levels, and data from the national mapping authorities of Canada, Denmark, Finland, Iceland, Norway, Russia, Sweden and USA. The service has been designed for use within the ArcticSDI Geoportal.
Indigenous peoples of the Arctic counties
Indigenous peoples of the Arctic countries subdivision according to language families. This service is internal and only used in the Arctic-SDI geoportal: https://geoportal.arctic-sdi.org/

redpajama_set_name: RedPajamaCommonCrawl


Luxury Golf Guide: 14 Great gifts that only the keenest golfers will want at Christmas this year.
Holing those crucial six-footers can really lower your scores, and the key to great putting is practice. With PuttOUT you really can take that practice anywhere. With a scientifically shaped parabolic ramp, each putt is returned to the player the same distance it would have gone past the hole had it missed; anything else is a miss. Only putts with perfect line and pace will hold the microtarget – so let the allimportant putting practice begin!
RRP: £169 (special offer from November 19 until December 23).
Shot Scope started out in 2014 and has set the standard for stat-based game improvement. The V2 utilises a GPS watch, to give distances on your wrist, but also record where shots go. You get access to more than 100 tour-level stats, and we've found Shot Scope has one of the most useful analysis platforms around. You can drill down into interactive graphs and charts to learn everything about your tee shots, approach shots (outside 50 yards), short game (inside 50 yards) and putting. As the system includes a GPS watch, it's really good value, too.
The grips seamlessly connect any set of clubs to the award-winning performance tracking system - the Arccos Caddie 2.0, powered by the Microsoft Azure cloud. The Smart Grips feature free access to golf's Smartest Caddie, powered by Artificial Intelligence that records and analyses every shot. The Arccos app, allows you to review distance and performance data, and receive live 'Plays Like' distances on the course and Tour Analytics to help make smarter decisions. Set includes 13 Lamkin Crossline 360 Smart Grips and one Arccos putter sensor.
RRP: £379.95, including 12-month Birdie Membership.
Everyone loves a gadget at Christmas, and SkyCaddie's feature-packed, top-of-the-range SX500 is one of the very best. It boasts striking new PinPoint Technology, which is tempting habitual laser users back to GPS. Yardages to the pin, with a GPS – whatever next? A huge five-inch HD glove-friendly screen, 35,000 pre-loaded exclusive SkyCaddie course maps, WiFi, rugged design and extended battery life make this one of this year's must-have golfing gadgets – from the brand that invented golf GPS.
Ideal for combating the coldest of temperatures, the ZeroFit Heatrub Ultimate is twice as warm as a jumper and yet is thin and lightweight enabling complete ease of movement for active sports. Unlike a standard base layer the fibres on the inside of the Heatrub Ultimate rub against the skin to create instant frictional heat, making it feel warm the instant you put it on. Available in red, charcoal, grey or black, in sizes S-XL.
Turn your TV or mobile device into a golf course for a year-round golfing experience in the comfort of your lounge or office. PHIGolf is a game simulator that attaches to the end of a training stick or regular club. It immerses you in a realistic round on a number of pre-installed courses, playable on various devices including smartphones, tablets and HD TVs. There are seven modes – from 18-holes to nearest the pin and putting comps. Introduce your family and friends to golf with the perfect Christmas gift.
RRP: £49.99 (Today's Golfer readers get 20% off using TGXMAS18 (offer available on orders placed before December 31, 2018. Delivery charges apply).
PurePutt from GrooveFix is the first adjustable putting gate on the market. Set the PurePutt at any measure between 50mm (1.97in) and 108mm (4.25in – the size of a regulation hole) and place on your chosen line. Practise putting through it from varying distances to get feedback on the club face angle, quality of strike at impact and putting accuracy. Packaged in a presentation pouch.
Leica is world-renowned for its high-quality optics, which have more than a century of development behind them. The crystal-clear Pinmaster II Pro is packed with features, especially Angle Compensated Distance (ACD) Technology. It takes sloping terrain into account to provide a second distance corrected for the slope (you can switch modes for competition play). First-Target Logic gives you a signal when you've locked onto your target, rather than an object behind it, and it's all packaged in an eyecatching, high-quality white body.
Do you struggle to read greens? Are you forever dropping shots because you don't play enough break? Golf Scope is an augmented reality green-reading app that unlocks the power of your iPhone to help you save shots on the green. It uses your phone's camera to read the topography of a green, and identify the break range and ideal pace of a putt. It then generates a trajectory for the optimum putt, and overlays that on your phone's view of the green. It also offers tips on where to aim and how hard to hit the ball; You can track your progress and spot trends as you accumulate putts; Video your stroke to analyse it later.
The all-new Golfstream Vision has been developed to provide the very best user experience available in an electric golf trolley today. Golfers told Golfstream they were fed up with screens that couldn't be seen in full sun... that tyres wore out in next to no time... that putting a trolley together was too complicated and that their bag twisted and wobbled. So their designers improved all these features. Buy a Vision plus lithium batter and get a waterproof cart bag for FREE!
Every golfer wants to hit it straighter. And every golfer wants to hit it further. SWINGYDE will help you do both, and you can use it at home or on the range. For less than £25 it's a lightweight, portable training aid tool we could all benefit from using. It will help you: Groove the correct swing positions; Set the correct hinge at the top of your backswing; Fix your clubface alignment throughout your swing. It is suitable for both right and left-handed men, women and junior golfers.
The revolutionary tracking system from Arccos Golf features 14 sensors – one for each club - that automatically record and analyse every shot you hit during a round. You're then able to review distance and performance data on the Arccos app. By providing golfers with more optimum club and strategy choices, the Arccos Caddie Smart Sensors help amateur golfers lower their handicaps. In 2017, Arccos users lowered their handicap by an average of 3.55 strokes, representing a 46x greater rate of improvement than the average golfer.
Normal RRP of £445, for a superb £245 while stocks last.
With lithium batteries now the norm in the majority of electric trolleys, Top Caddy is your one-stop shop for all lithium requirements. Whether it's a trolley battery that lasts for 27 holes (£139), 36 holes (£149) or a lithium battery for a golf buggy, you'll find it here. They also offer industry-leading 24v extended mileage lithium packages, with free buggy connectors and free shipping (£575). Top Caddy is also offering the Golfstream Vision digital lithium trolley.
GoKart has always been a popular, great value British-made electric trolley. But for 2019, they've made it even better. The first thing you'll notice is the smart new styling. A unique synchronised folding frame now has an auto lock, so when you're getting ready the trolley just clicks into place without any messing. A new lithium battery slots in and connects without any fiddling and you can charge the battery when it's still in place. Other changes under the skin include improved weight distribution to makes it more stable plus a powerful new motor.


text:
You are here: Understand Tenant Dial Plans > Types of Dial Plans
Types of Dial Plans
Virtual Contact Center offers two types of dial plans:
System Pre-Configured Dial Plans: Predefined national and international dial plans available to all tenants. They allow PBX dialing and local, long distance, and international calls. You cannot make any changes to these plans except for opting a default plan. You can select a national plan that best suits your needs or use the standard international plan. The System Pre-Configured dial plans are predefined and do not allow changes. Every tenant is required to select one and only one dial plan to suit their location. By default, your contact center may come with some system pre-configured dial plans, such as:
International Numbering Plan (ITU-T E.164): Allows tenants across the globe to place domestic and international calls. The International Numbering Plan does not allow unfiltered pass-through dialing.
US North American Numbering Plan (US NANP): Allows tenants across North America to place domestic and international calls. NANP identifies various calling number patterns and applies corresponding treatment to user-dialed numbers.
Custom Dial Plans: In addition to system pre-configured dial plans, you can create your own custom dial plan if you have subscribed to this feature. Custom dial plans, in many situations, can mimic the abbreviated dialing conventions of a private office phone system or internal corporate dialing plan. You can configure multiple custom dial plans.
-You can activate only one dial plan per tenant at a time.
-Virtual Contact Center reports present dialed numbers in the standard international format.
About the North American Numbering Plan (US NANP)
The North American Numbering Plan identifies various calling number patterns and applies the following treatment to user-dialed numbers:
Filters international calls with country codes incorrectly entered starting from "0".
Note: Country codes do not begin with 0.
Removes the North American access code 011 if added to a North American phone number. For example, the number 01116505551212 is dialed out as 16505551212, removing 011.
Filters local and long-distance calls with incorrect area codes starting from "0".
Note: North American area codes do not begin with 0.
Filters calls to emergency and special services, such as 911 in the US and 112 in the UK.
Prefixes the country code "1" to all local and long-distance calls within North America.
Requires country codes and area codes for international calls.
Allows input of spelled-out numbers, and converts alphabetic characters to the telephone keypad equivalent digits according to North American Telephone standards.


The cool off-season is upon us and we could not be more excited. Owning a Cal Spas Hot Tub will enhance your life this holiday season by bringing warmth to your backyard oasis. Fall is the season where you gather next to the heat and drink hot cocoa and eat delicious sugary desert. It is also the season of gathering and events. Let the party continue all season long with Cal Spas!
So you may ask, how does Fall enhance my hot tub?
A great way to create memorable activities is to create a focal point on spending time together in your Cal Spas Hot Tub or Swim Spa. Turn on the C-Light, set up your favorite playlist, add aromatherapy, and create the best night for your friends and family to truly enjoy outside.
Planning parties and getting ready for holiday festivities are always stressful, but yet super fun. A great way to unwind from festivities is having your friends and family unwind with you in a Cal Spas Hot Tub or Swim Spa. Even better, when the party is over you are able to escape into your outdoor oasis and rejuvenate for the next day.
Winterize your Hot Tub Winterizing your spa for the season draining it, cleaning it, and protecting it from the effects of freezing weather can be a complicated and time-consuming procedure. In many cases, it's better for the health of your hot tub to keep it running. Your spa will not freeze overnight unless the water stops circulating and you live in -30° F. If your spa looses power, and assuming there is a cover with reasonable insulation, you will have at least a few days before water starts to freeze. In fact, keeping up your spa maintenance in winter has special advantages; follow a few tips to keep your spa in prime condition. Here are 5 maintenance tips to keep you up to date and your spa running smoothing all winter long.


Cotic, the Peak-district based purveyors of fabricated out of animate bicycles aboriginal alien the 29in/27.5 wheeled FlareMAX archetypal in 2016.
This is the additional abundance of Cotic's 120mm travel, do-it-all, up-for-most-things, abounding abeyance aisle bike. While at aboriginal glance it looks accustomed to the aboriginal adaptation there accept been abundant attenuate changes to fixtures, fastenings, the agreement of things, architecture detail and best conspicuously the unleashing of Longshot geometry.
What's that then? It is, of course, the movement of the geometry slider appear slacker and longer. Don't blow too much, there's a bit added to it than that.
Cotic has been experimenting with accelerating geometry for a while now. We aboriginal saw it with a collaboratively congenital ancestor bike dubbed the 'Longshot' aback in 2016 at the Bristol Bespoked bike show. This new geometry cerebration is now a key allotment of how Cotic do things. Cy and his amusing aggregation are designing all of their new address abundance bikes the Longshot way.


Over the long winter, Julep has matured and blossomed into a swan. Her elegance abounds and her movement extends to a new level.
Julep has remained steadfast in her most people-pleasing temperament.
A visitor to the farm wanted to see just how cooperative a Belfrey's baby was. So I pulled her out of her paddock to the driveway where she had never been before.
There was a heap of mulch piled 5 feet high with a vertical sheer where we had been moving it shovel by shovel.
heap. Julep went right to the heap, sniffed it and elevated her extremely elastic shoulder and climbed vertically up and over the heap!
While on the other side, I pointed for her to come back across the mountain. She climbed up and then confidently slid down the 5ft. vertical drop onto the asphalt. Her visitors were thoroughly impressed!
When I pointed to their car...and she lifted her shoulder... they exclaimed, "NO, NOT THE CAR!!" They were convinced beyond a shadow of a doubt at the willingness of Belfrey's foals!
all! Incredible temperament, balanced conformation (fantastic shoulder, powerful haunch, amazing topline, graceful neck and straight limbs), superior athleticism and elastic movement all in one gorgeous package!
Julep IS as sweet as she looks!
As a last minute decision, we took our yearling filly, BenMar Mint Julep, IDSH to her first show, the May USDF KDA Breeders Classic I & ll. I was politely told that I was crazy to introduce clipping, bathing, and trailering the same day we left for the show, but I just didn't think it would be a problem.
And it wasn't. Julep didn't even look twice at the clippers. Her head remained low and relaxed. Bathing was a non-issue. It was time to load up for the first time. I led Julep to the trailer, pointed my finger and said "Walk on." I threw the rope over her back and she loaded herself!!! Julep made an indelible first impression for many of the top Warmblood breeders and German and American Dressage Show Judges at the Lexington Horse Park, KY. A small crowd gathered from the moment she walked off the trailer. In a sea of fine Warmbloods, Julep outperformed her counterparts by off-loading herself from the trailer with the total peacefulness of a well-seasoned show horse. With relaxed curiosity, she absorbed the busy show atmosphere without moving a foot. Julep's audience was amazed when they heard this was her first time loading, trailering and off-loading.
We were stunned after her first class. The German Judge for the Open Yearling Filly Class placed Julep in the middle out of 9 entries against all Warmbloods specifically bred for dressage! Julep then took first place and the Championship for Irish Draught PartBreds over a mature stallion and mare! The two judges loved her "elegant head, nice type, well developed shoulder and hindquarters, and good rhythm." The judges were impressed with Julep's movement in spite of her growth spurt and remarked that we will see dramatic improvements as she grows. Her professional handler, Bruce Griffin, was so pleased with Julep's willing demeanor that he repeated, "This is the kind of horse that you want to ride!" Bred for versatility and not specifically for dressage, we were overjoyed with her accomplishments.
Julep's wins were secondary to her noteworthy representation of the Irish Draught Sport Horse and her Champion Sire, Bridon Belfrey, RID. Julep's calm, sensible approach to vast changes in her environment compelled remarks such as: "There is no way my horse would ever be this good. I can't even braid my filly in her stall for fear she will kill me!" and "Looks like she stands quietly for the clippers." Julep stood patiently for 2 hours while I fumbled with her braids. They couldn't believe she was clipped for the first time that morning. After the show, with a point of the finger and a "walk on", she loaded herself and stood patiently while I closed the ramp. Many of her admirers had never met an Irish Draught Sport Horse, but now will never forget it!
Julep is already exceeding our expectations with her brave and comfortable approach to new challenges. Her daily routine includes navigation a myriad of obstacles, many of which are in a dark, narrow barn isleway. With a point of a finger, she eagerly walks 12 feet in front of me, down our 3 foot wide barn isle, UNDER a ladder while simultaneously climbing several haphazard bales of hay! She adsolutely LOVES the adventure!
She leads from either side equally well. With a point of a finger she will adjust her speed. She trots when I run and stops instantly without touching her. She is by my side wherever I go. I take her on new adventures everyday. We often visit our water fountain, the mailbox and pallets of mulch and concrete (whatever we happen to have delivered). One day we decided to visit my husband while he was on a conference call. We walked right up our sidewalk through the courtyard, to the front porch, up two steps, ON THE FRONT PORCH and rang the doorbell!!! My husband came out and was shocked to see Julep's innocent face and bright eyes with eye lashes just a batting! He said to the 15 people on the conference call, "Uh, can you hold on for a second!?" After loving on Julep, he went back to his meeting explaining what just happened! Hahahaha! It was PRICELESS!
We are pleased to announce the much anticipated arrival of our newest sport horse foal! Julep is of outstanding quality and elegance and has the most exceptional temperament of all the foals produced at BenMar Farm.
At 2 hours old, she literally walked to our daughter, leaned against her and remained there for all her hugs and scratches! We've never seen such a sweet foal! Julep has learned to lead confidently in only one brief lesson, showing a great aptitude and willingness for training. She is already an athletic mover displaying Belfrey's trademark elevated canter and forward trot. She inherited Belfrey's exceptional temperament, topline, powerful haunches and great shoulder angle. We expect Julep will make a fantastic all around family horse that will excel in Fancy Hunters, eventing, dressage, driving or hunting.
Julep's super athleticism and sweet, giving temperament gives her the versatility to excel in any discipline. We feel she has the potential of achieving national level rankings in Hunters, eventing or showjumping.
Check back for new photos often!


text:
DIARY OF AN EXPAT WRITER: A tale of two names, Shannon Young (expat writer) and Jordan Rivet (hipster expat writer)
1 Comment Posted by The Displaced Nation Team on April 6, 2016
The last time we heard from American expat in Hong Kong and aspiring writer Shannon Young, she was hatching some big plans for 2016. So, how many books has she managed to crank out since? Prepare to be amazed…!
Dear Displaced Diary,
As you may recall, two months ago I laid out my goals for the beginning of 2016:
Finish and publish Ferry Tale, a love story set in Hong Kong, under the name Shannon Young.
Compile and publish a box set of the Seabound trilogy.
Finish and publish Duel of Fire, the first book in my new fantasy series, under the name Jordan Rivet.
I'm happy to report I'm right on track:
#1 Ferry Tale launched in ebook and paperback just in time for Valentine's Day. It's a classic romantic comedy with a modern twist, featuring an American woman and a Chinese Canadian man. Here's the cover, designed by James at GoOnWrite.com:
#2 The Complete Seabound Trilogy Box Set launched in March, along with one of my most successful promotions to date. This is a compilation of the post-apocalyptic adventure set at sea that I released over the course of 13 months, starting in November 2014. (I've reported about my progress on this in previous diary entries.)
#3 Duel of Fire is finished, edited, polished, and formatted. It went on sale last Friday, April 1st, right on schedule (no fooling!). Here is the cover from Deranged Doctor Design. PLEASE NOTE: It costs $0.99 for the first week, so definitely pick up a copy SOON if you like sword fights, magic, adventure, and a hint of romance!
I'm currently hard at work on the sequel, King of Mist, which will come out in May.
What's in a (pen) name, really?
You might notice that these books are published under two different names. Shannon Young is the name given to me at birth, and I picked my pseudonym, Jordan Rivet, myself.
Have you ever wondered about that, Dear Diary? Were you thinking I might have a dual personality?
The simple answer as to why I write under two names is branding. Looking at those two book covers, which hopefully communicate their genres at a glance, you'd never know the same person wrote the books. They are not similar in any way, so you would never scan past Duel of Fire in a listing of books on Amazon and think: "It's another Shannon Young book! I have to stop and click!"
You probably wouldn't even be looking at the same lists. Although some readers enjoy both epic fantasies and romantic comedies (I like to read both, after all), for the most part these books are targeted to distinct groups.
To some, I am known as Shannon Young
The Shannon Young "brand" is already quite eclectic. It includes a bit of travel, a bit of romance, some stuff about being an expat, and some stuff about being a millennial. These bits and pieces are a reflection of my writing journey—and myself as a person. However, they haven't established me as a type of author in the way that, say Nicholas Sparks, Jodi Picoult, or Malcolm Gladwell are understood to represent a particular type of book.
Indeed, although my Shannon Young work represents me as a growing and changing person, my first love as a reader has always been fantasy and science fiction.
Diary, I seem to be evolving in the direction of a Nora Roberts. Roberts is of course known for one type of fiction (romance), whereas her pseudonym, JD Robb, is famous for another type (futuristic scifi police procedurals). Except that I'm approaching it the other way around: building a brand around my pseudonym first, as my Jordan Rivet books outsell my Shannon Young books by a significant margin. Launching the Jordan Rivet name meant launching an income-generating career.
Enter: Jordan Rivet
When I started the Seabound Chronicles, a post-apocalyptic adventure set at sea, I already knew it wouldn't match my existing books. My readers wouldn't necessarily follow me quite that far across a genre divide. I also knew I'd likely write some high fantasy when I became a good enough writer to try my hand at my very favorite genre of all. Since I'd be starting from scratch with my readership, I figured I'd go the whole way and create a new Science Fiction & Fantasy (SFF) brand.
The Jordan Rivet name has never been a secret identity (unlike JK Rowling's Robert Galbraith), but I had fun choosing the name and thinking about how to present the brand. It was a creative exercise just like developing a character for a novel. I wanted a name that was:
gender ambiguous (hello, scifi readers!)
uncommon enough that I'd no longer be confused with all the other Youngs out there (they are legion).
And the name I settled on even pays homage to one of my favorite fantasy writers (RIP Robert Jordan).
I like to think of Jordan Rivet as the cooler, sexier version of myself. She works harder, writes faster, is better at writing dialogue, and takes more risks than the more introspective Shannon Young.
You expats will likely recognize this split-personality complex. You probably feel like a different version of yourself depending on which country you are in at any given time. And you know what it's like to resume an old role when you return to your home country for a visit and pick up right where you left off as soon as the holiday ends.
Surprisingly, it's possible to do this as a writer as well.
Two publishing paths diverged in a narrow . . . hipster neighborhood?
Diary, as you may have gleaned from my previous entries, there's a further way Shannon Young and Jordan Rivet differ: in their publishing paths. I've only ever queried agents and publishers and signed traditional book deals as Shannon Young. It's very much a hybrid name and includes some small press titles, an audiobook deal, a Kindle Single, and some self-published work. As Shannon Young, I've been able to speak at the Hong Kong International Literary Festival and present the traditionally published "stamp of legitimacy" that some outlets still require.
But Jordan Rivet is an indie name all the way. I've never submitted a Jordan Rivet story to an agent. I publish the books when they are edited and ready, not when someone tells me I can. It has been incredibly empowering to take this path, and it has prompted me to improve my writing and to pay more attention to the market.
And you know something, Diary, I have a good feeling about Duel of Fire, a book that wouldn't exist yet if I were still waiting for the Seabound series to make its way through the traditional pipeline.
Obviously, different methods are right for different books and different types of writers. But indie publishing has allowed me to take my work to the next level, and I can't wait to see the outcome of the launch that began this past weekend. Don't forget to pick up your discounted copy pronto!
So, Diary, that is the story of why I write under two different names. When people ask me what I write in real life, small talk gets a bit complicated—a bit like when a Third Culture Kid being asked where they're from! (My author pages on Amazon make a lot more sense!)
AKA Jordan Rivet
p.s. to Diary Readers: Do you have experiences you can share where you've adopted different personas in different situations? Do these personas come with different names too?
To answer your question, Shannon, I definitely think we expats and former expats are candidates for multiple names. We're after all the kind of people who like to live out our fantasies, and some of us even write about them. Thanks for being one of them, Jordan! 🙂 ~ML
STAY TUNED for more fab posts!
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DIARY OF AN EXPAT WRITER: Back in full-time writing mode and full of resolution(s) for 2016!
DIARY OF AN EXPAT WRITER: Still reeling from reaching the end of my 4-book series in 3 years!
DIARY OF AN EXPAT WRITER: The highs, but also the lows, of the writing life
DIARY OF AN EXPAT WRITER: How I spent my summer vacation
DIARY OF AN EXPAT WRITER: How far I have come in a year…and what's next
DIARY OF AN EXPAT WRITER: Three expat writers walk into a bar . . . and are giving away their books (limited time only!)
DIARY OF AN EXPAT WRITER: After the hard slog of producing second book in series, the rush that comes with boost to sales
DIARY OF AN EXPAT WRITER: My writing process, now being applied to a shiny new book series
DIARY OF AN EXPAT WRITER: Shifting base to literary London, where it all began
DIARY OF AN EXPAT WRITER: Has it really been six months? How time flies when you're writing full time
DIARY OF AN EXPAT WRITER: There's no place like home, especially when your compatriots are reading your books!
DIARY OF AN EXPAT WRITER: Will she, or won't she, on Christmas Day? (We're taking bets!)
DIARY OF AN EXPAT WRITER: Turning into Jordan Rivet, writer of post-apocalyptic adventures, for an entire month
DIARY OF AN EXPAT WRITER: How to fly solo without burning the tips of my wings
DIARY OF AN EXPAT WRITER: Testing one, two, three…Can anybody hear me?
Photo credits: Photos of Shannon and her books were supplied.
Cool Columns, Diary of an Expat Writer China, Diary of an Expat Writer, fantasy writers, Hong Kong, It's Fiction!, Love love love, Memoir | Travel | Food Writers, Third Culture Kid, USA
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1. This pretty sandy beach means 'stones cove' in Cornish. It's sheltered in a wooded valley, soaked in sunshine, and is just two miles from Falmouth town.
2. It offers everything you need for the ultimate beach day – sun bathing, snorkelling, fishing, kayaking, toe-dipping and boating.
4. Or you can park directly behind the beach, which is £3 per visit during summer.
5. Maenporth is in a prime location with panoramic views across Falmouth Bay and Pendennis Castle, plus good coastal walks onwards towards the Helford.
6. Nature enthusiast? Try rock pooling in its shallow waters or bring your binoculars. It's a rich hub for sea critters and birdlife – from starfish and urchins to herons and egrets.
7. Life's a Beach is Maenporth's café – with a chilled out atmosphere, it's a great retreat to savour a fresh crab sandwich or homemade soup if it's chilly.
8. If you forget your bucket, spade, fishing lines or even a wetsuit, the friendly folk here can kit you out.
9. If the tide is low, you can follow the rocks down the left hand side of the cove and might be lucky enough to find the famous shipwreck of the Ben Asdale enveloped in the sand.
10. There are regular yoga sessions on the beach ran by a lovely lady called Naomi if you fancy finding your inner calm.
Or head to the beach at sunrise or sunset to try SUP Yoga on a paddle board.


Who made a strong showing at the GOP Iowa debate yesterday?
The Ames Straw Poll, arguably the first major vote of the 2012 presidential campaign, is set to happen this Saturday. The poll is not binding, but three out of five times the winner has gone on to win the Iowa caucus and the nomination.
As a lead up to the event, last night, Republican presidential candidates squared off at a debate sponsored by Fox News and The Washington Examiner and held in an arena at Iowa State University.
Today on American Morning, Al Diguido, CEO of ZETA Interactive, and Ann Selzer, independent pollster and owner of Selzer & Company, join Carol Costello to weigh in on who made a strong showing at last night's debate and who is the front runner heading into this weekend's straw poll.


text:
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Land Size To (acres) 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760
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What began as learning an essential life skill, became the very thing that defined my life.
While my passion for swimming is able to bring success and has inspired many, I knew that I wanted to do something more; I wanted to give every child the same opportunity that I had, to dream bigger.
Whether it is about building confidence in the water or about gliding towards your aspirations and goals; Your journey begins with you.
We are a swim school that believes in empowering every child's aquatic dream. We teach, develop and nurture each individual to bring out their best through our unique rewards programme, small class numbers and the experience and credibility of our teachers, who aim to inspire.


Autumn-Winter 2016 could see a lot of velvet and in every form. Pants, shirts, blouses, bodycon, skirts, shorts to thigh high boots, it was a fabric used almost everywhere.We can never get enough of velvet. This time from casual to party wears, velvet caught eyes of many. The best part is it wasn't even expensive as fur and easy to be managed.
The websites are full of it and the stores were loathed with the velvet collection. I also managed to get few key pieces and wore them not only in darkhours but also in daytime. Still can't get over it. Choosing the right colors, this can be your next satin to officewear closet.
Styling this fabric is a little tricky but not as hard as you may think. Some find this royal fabric a little too much to wear during days and would rather keep it in their party closet. But not anymore, you can always dress it down by choosing the right add-ons to it. Layer it, if you can't carry it alone. Otherwise, don't be afraid to go for lighter fabrics when choosing the top or bottom as velvet.
And who can resist the allure of velvet? It's a fabric of nobility and royalty (Princess Diana was a fan) and on the other end of the spectrum, it boasts a dangerous streak, with connections to glam rock and nighttime seduction. And now, the smooth, luxurious material has a wonderfully arty-eccentric offbeat side—only one element of what makes up the "Gucci girl" that creative director Alessandro Michele dreamed up (see: the fall/winter 2016 collection) Trending colors in velvet this season are royal blue, shades of red, violets, green and marsala.
Look at the bodycon I picked up from STALKBUYLOVE and styled it my way for my day look. Same bodycon can be used with denim, skirts for a more playful look which you can find on my INSTAGRAM (@indianchic.in).
Go ahead! Rock your velvet this season.


iDEC 5S Practice and Programme Committee will implement Gotong-Royong 2018. The Gotong-royong was conducted in preparation for the 5S Practices and Programming Audit to be held in August 2018 to win the Workplace Environmental Quality Award. As a courtesy, the 5S Practical Appraisal and Practice Audit scores also affect the scoring for the Star Rating Management Rating Awards.
To that end, all iDEC staff are invited and asked to attend briefings and participate in mutual-aid activities. All 5S Zone Leaders (Alpha Builders, Gamma, Sigma, Beta) are required to notify this activity to all staff in their respective buildings.
Further information and programs are as in the attached poster. The smoothness and success of this activity depends on the cooperation and commitment of all iDEC staff. Thanks to the cooperation, it is preceded by thanks.


Building one of the dozen fastest supercomputers isn't new for Cray – they've got three in the current top 12 now. But what is unique is that most of those 12 belong to government research labs or universities, not private companies. This may be starting to change, however. For example, IDC notes that overall supercomputing spending in the oil and gas sector alone is expected to reach $2 billion in the period from 2013-2018.
In the 1990s I was involved in a student project to build a Linux Beowulf cluster out of commodity components. It involved a half a dozen quad-core servers, with something like a gigabyte of RAM each. It cost a fortune, and it required us to obtain NSF funding for the project. I don't recall the exact details.
I know, that a similarly configured modern cluster in AWS would cost a few hundred bucks a month if it was used continuously. But even the cluster we built at Clarkson was not used 24/7, and so if done right the same cluster would have cost a fraction of that in AWS.
That last statement is important. Setting aside the costs, imagine the red tape required to put something like that together with the help of your on-premise IT department ?
Bristol-Myers Squibb (BMS) is a global biopharmaceutical company committed to discovering, developing and delivering innovative medicines that help patients prevail over serious diseases. BMS used AWS to build a secure, self-provisioning portal for hosting research so scientists can run clinical trial simulations on-demand while BMS is able to establish rules that keep compute costs low. Compute-intensive clinical trial simulations that previously took 60 hours are finished in only 1.2 hours on the AWS Cloud. Running simulations 98% faster has led to more efficient and less costly clinical trials and better conditions for patients.
If I interpret that case study correctly BMS didn't even bother with an on-premise supercomputer for this.
AWS allows you to increase the speed of research by running high performance computing in the cloud and to reduce costs by providing Cluster Compute or Cluster GPU servers on-demand without large capital investments. You have access to a full-bisection, high bandwidth network for tightly-coupled, IO-intensive workloads, which enables you to scale out across thousands of cores for throughput-oriented applications.
So, what would it cost to setup one of the worlds most powerful supercomputers in AWS and run it for one month ? I fully realize that this may not be a very accurate discussion, but let's humor ourselves and try to imagine the biggest of the Top 500 Supercomputers built in AWS.
As of June 2013, the biggest super computer was at National University of Defense Technology in China, and it had 3,120,000 CPU cores. Let's eyeball this in AWS using Amazon's cost calculator. I put together a coupe of different HPC configurations.
Amazon's g2.2xlarge instances have 8 cores and 15 gigabytes of RAM each. To get to the 3,120,000 cores one would need 390000 instances, which would cost $185,562,000.00 for a month, not including business support.
If you use No-Upfront Reserved for 1 year, the cost becomes $134,947,800.00 per month for a year. Three Year All-Upfront Reserved costs $2,889,900,000.00 up front and $100 a month.
Now, here is an important factor. On premises you have to build out the maximum capacity you will ever use, but in the cloud you can dynamically scale up and down as required by your workload. Whereas supercomputing was the domain of governments and wealthy corporations, it is now within reach of anyone building out in AWS.
Let's try this with c4.8xlarge. On-demand this costs $119,901,600.00 a month. Three Year All-Upfront is $1,609,335,000.00 .
Of course, we don't even know if such a thing is even possible on AWS — to quickly spin up a few hundred thousand servers. How long would that take ? This would probably require a conversation with AWS sales, and probably a volume discount. But either way, something tells me that for such large specialized computational workloads it would be naive to assume that building a supercomputer in the cloud would be cheaper.
This is why renting supercomputing time is still more efficient than both owning one or trying to spin one up in the cloud.