Private Information Retrieval From a Cellular Network With Caching at the Edge

File $\bm{X}^{(i)}$ is partitioned into $\beta k_{i}$ packets of size $L/k_{i}$ bits and encoded before being cached in the SBSs. In particular, each packet is mapped onto a symbol of the field $\mathrm{GF}(q^{\delta_{i}})$, with $\delta_{i}\geq\frac{L}{k_{i}\log_{2}q}$. For simplicity, we assume that $\frac{L}{k_{i}\log_{2}q}$ is integer and set $\delta_{i}=\frac{L}{k_{i}\log_{2}q}$. Thus, stripe $\tilde{\bm{x}}^{(i)}_{a}$ can be equivalently represented by a stripe $\bm{x}^{(i)}_{a}$, $a=1,\ldots,\beta$, of symbols over $\mathrm{GF}(q^{\delta_{i}})$. Each stripe $\bm{x}^{(i)}_{a}$ is then encoded using an $(N_{\mathsf{SBS}},k_{i})$ MDS code $\mathcal{C}_{i}$ over $\mathrm{GF}(q)$ into a codeword $\bm{c}^{(i)}_{a}=(c^{(i)}_{a,1},\ldots,c^{(i)}_{a,N_{\mathsf{SBS}}})$, where code symbols $c^{(i)}_{a,j}$, $j=1,\ldots,N_{\mathsf{SBS}}$, are over $\mathrm{GF}(q^{\delta_{i}})$. For later use, we define $k_{\mathsf{min}}\triangleq\min\{k_{i}\}$, $k_{\mathsf{max}}\triangleq\max\{k_{i}\}$, and $\delta_{\mathsf{max}}\triangleq\frac{L}{k_{\mathsf{min}}\log_{2}q}$.

The encoded file can be represented by a $\beta\times N_{\mathsf{SBS}}$ matrix $\bm{C}^{(i)}=(c^{(i)}_{a,j})$. Code symbols $c^{(i)}_{a,j}$ are then stored in the $j$-th SBS (the ordering is unimportant). Thus, for each file $\bm{X}^{(i)}$, each SBS caches one coded symbol of each stripe of the file, i.e., a fraction $\mu_{i}=1/k_{i}$ of the $i$-th file. As $k_{i}\in\{1,\ldots,N_{\mathsf{SBS}}-1\}$,

where $\mu_{i}=0$ implies that file $\bm{X}^{(i)}$ is not cached. Note that, to achieve privacy, $k_{i}<N_{\mathsf{SBS}}$, i.e., files need to be cached with redundancy. As a result, $\mu_{i}=1/N_{\mathsf{SBS}}$ is not allowed. This is in contrast to the case of no PIR, where $k_{i}=N_{\mathsf{SBS}}$ (and hence $\mu_{i}=1/N_{\mathsf{SBS}}$) is possible.

Since each SBS can cache the equivalent of $M$ files, the $\mu_{i}$’s must satisfy

We define the vector $\bm{\mu}=(\mu_{1},\ldots,\mu_{F})$ and refer to it as the content placement. Also, we denote by $\mathcal{C}_{\mathsf{MDS}}^{\bm{\mu}}$ the caching scheme that uses MDS codes $\{\mathcal{C}_{i}\}$ according to the content placement $\bm{\mu}$. For later use, we define $\mu_{\mathsf{min}}\triangleq\min\{\mu_{i}|\mu_{i}\neq 0\}$ and $\mu_{\mathsf{max}}\triangleq\max\{\mu_{i}\}$.

We remark that the content placement above is slightly different than the content placement proposed in [7]. In particular, we assume fixed code length (equal to the number of SBSs, $N_{\mathsf{SBS}}$) and variable $k_{i}$, such that, for each file cached, each SBS caches a single symbol from each stripe of the file. In [7], the content placement is done by first dividing each file into $k$ symbols and encoding them using an $(\tilde{n}_{i},k)$ MDS code, where $\tilde{n}_{i}=k+(N_{\mathsf{SBS}}-1)m_{i}$, $m_{i}\leq k$. Then, $m_{i}$ (different) symbols of the $i$-th file are stored in each SBS and the MBS stores $k-m_{i}$ symbols.²²2This is because the model in [7] assumes that one SBS is always accessible to the user. If this is not the case, the MBS must store all $k$ symbols of the file. Here, we consider the case where the MBS must store all $k$ symbols because it is a bit more general. Our formulation is perhaps a bit simpler and more natural from a coding perspective. Furthermore, we will show in Section IV that the proposed content placement is equivalent to the one in [7], in the sense that it yields the same average backhaul rate.

Mobile devices request files according to the popularity distribution $\bm{p}=(p_{1},\ldots,p_{F})$. Without loss of generality, we assume $p_{1}\geq p_{2}\geq\ldots\geq p_{F}$. The user request is initially served by the SBSs within communication range. We denote by $\gamma_{b}$ the probability that the user is served by $b$ SBSs and define $\bm{\gamma}=(\gamma_{0},\ldots,\gamma_{N_{\mathsf{SBS}}})$. If the user is not able to completely retrieve $\bm{X}^{(i)}$ from the SBSs, the additional required symbols are fetched from the MBS. Using the terminology in [7], the average fraction of files that are downloaded from the MBS is referred to as the backhaul rate, denoted by R, and defined as

Note that for the case of no caching $\textnormal{R}=1$.

As in [7], we assume that the communication is error free.

We assume that some of the SBSs are spy nodes that (potentially) collaborate with each other. On the other hand, we assume that the MBS can be trusted. The users wish to retrieve files from the cellular network, but do not want the spy nodes to learn any information about which file is requested by the user. The goal is to retrieve data from the network privately while minimizing the use of the backhaul link, i.e., while minimizing R. Thus, the goal is to optimize the content placement $\bm{\mu}$ to minimize R.

The queries must be constructed such that privacy is preserved and the user can retrieve the requested file from the $n$ response vectors $\bm{r}^{(l)}$, $l=1,\ldots,n$. In particular, the protocol is designed such that the subresponses $r^{(l)}_{j}$, $l=1,\ldots,n$, corresponding to the $n$ subqueries $\bm{q}_{j}^{(1)},\ldots,\bm{q}_{j}^{(n)}$ recover $\Gamma$ unique code symbols of the file $\bm{X}^{(i)}$.

The queries are constructed as follows. The user chooses $\beta F$ codewords $\bar{\bm{c}}_{m}^{(i)}=(c_{m,1}^{(i)},\ldots,c_{m,n}^{(i)})\in\bar{\mathcal{C}}$, $m=1,\ldots,\beta$, $i=1,\ldots,F$, independently and uniformly at random. Then, the user constructs $n$ vectors,

where $\mathring{\bm{c}}^{(i)}_{l}$ collects the $l$-th coordinates of the $\beta$ codewords $\bar{\bm{c}}_{m}^{(i)}$, $m=1,\ldots,\beta$, i.e., $\mathring{\bm{c}}^{(i)}_{l}=(\bar{c}^{(i)}_{1,l},\ldots,\bar{c}^{(i)}_{\beta,l})$.

Assume that the user wants to retrieve file $\bm{X}^{(i)}$. Then, subquery $\bm{q}^{(l)}_{j}$ is constructed as

where

for some set $\mathcal{J}_{j}$ that will be defined below. Vector $\bm{\omega}_{t}$, $t=1,\ldots,\beta F$, denotes the $t$-th $(\beta F)$-dimensional unit vector, i.e., the length-$\beta F$ vector with a one in the $t$-th coordinate and zeroes in all other coordinates, and $\bm{\omega}_{0}$ the all-zero vector. The meaning of index $s_{j}^{(l)}$ will become apparent later.

According to (4), each subquery vector is the sum of two vectors, $\mathring{\bm{c}}_{l}$ and $\bm{\delta}_{j}^{(l)}$. The purpose of $\mathring{\bm{c}}_{l}$ is to make the subquery appear random and thus ensure privacy (i.e., Condition (2a)). On the other hand, the vectors $\bm{\delta}_{j}^{(l)}$ are deterministic vectors which must be properly constructed such that the user is able to retrieve the requested file from the response vectors (i.e., Condition (2b)). Similar to Protocol 3 in [21], the vectors $\bm{\delta}_{j}^{(l)}$ are constructed from a $d\times n$ binary matrix $\hat{\bm{E}}$ where each row represents a weight-$\Gamma$ erasure pattern that is correctable by $\tilde{\mathcal{C}}$ and where the weights of its columns are determined from $\beta$ information sets $\mathcal{I}_{m}$, $m=1,\ldots,\beta$, of $\mathcal{C}_{\mathsf{max}}^{\prime}$.

The construction of $\hat{\bm{E}}$ is addressed below. We define the set $\mathcal{F}_{l}$ as the index set of information sets $\mathcal{I}_{m}$ that contain the $l$-th coordinate of $\mathcal{C}_{\mathsf{max}}^{\prime}$, i.e., $\mathcal{F}_{l}=\{m:l\in\mathcal{I}_{m}\}$. To allow the user to recover the requested file from the response vectors, $\hat{\bm{E}}$ is constructed such that it satisfies the following conditions.

1. $\mathsf{C1.}$

   The user should be able to recover $\Gamma$ unique code symbols of the requested file $\bm{X}^{(i)}$ from the responses to each set of $n$ subqueries $\bm{q}^{(l)}_{j}$, $l=1,\ldots,n$. This is to say that each row of $\hat{\bm{E}}$ should have exactly $\Gamma$ ones. We denote by $\mathcal{J}_{j}$ the support of the $j$-th row of $\hat{\bm{E}}$.

2. $\mathsf{C2.}$

   The user should be able to recover $\Gamma d\geq\beta k_{i}$ unique code symbols of the requested file $\bm{X}^{(i)}$, at least $k_{i}$ symbols from each stripe. This means that each row $\hat{\bm{e}}_{j}=(\hat{e}_{j,1},\ldots,\hat{e}_{j,n})$, $j=1,\ldots,d$, of $\hat{\bm{E}}$ should correspond to an erasure pattern that is correctable by $\tilde{\mathcal{C}}$.

3. $\mathsf{C3.}$

   Let $\bm{t}_{l}$, $l=1,\ldots,n$, be the $l$-th column vector of $\hat{\bm{E}}$. The protocol should be able to recover $w_{\mathsf{H}}\left(\bm{t}_{l}\right)$ unique code symbols from the $l$-th response vector, which means that it is required that $w_{\mathsf{H}}\left(\bm{t}_{l}\right)=|{\mathcal{F}_{l}}|$. We call the vector $(w_{\mathsf{H}}\left(\bm{t}_{1}\right),\ldots,w_{\mathsf{H}}\left(\bm{t}_{n}\right))$ the column weight profile of $\hat{\bm{E}}$.

Finally, from $\hat{\bm{E}}$ we construct the vectors $\bm{\delta}_{j}^{(l)}$ in (5). In particular, index $s_{j}^{(l)}$ in (5) is such that $s_{j}^{(l)}\in\mathcal{F}_{l}$ and $s_{j}^{(l)}\neq s_{j^{\prime}}^{(l)}$ for $j\neq j^{\prime}$, $j,j^{\prime}=1,\ldots,d$.

The $j$-th subresponse corresponding to subquery $\bm{q}^{(l)}_{j}$, $j=1,\ldots,d$, is (see (1))

The user collects the $n$ subresponses $r^{(l)}_{j}$, $l=1,\ldots n$, in the vector $\bm{\rho}_{j}$,

where symbol $o^{(l)}_{j}$ represents the code symbol from file $\bm{X}^{(i)}$ downloaded in the $j$-th subresponse from the $l$-th response vector. Due to the structure of the queries obtained from $\hat{\bm{E}}$, the user retrieves $\Gamma$ code symbols from the set of $n$ subresponses to the $j$-th subqueries. Consider a retrieval code $\tilde{\mathcal{C}}$ of the form

where $\mathcal{C}^{\prime}_{i}+\mathcal{C}^{\prime}_{j}$ denotes the sum of subspaces $\mathcal{C}^{\prime}_{i}$ and $\mathcal{C}^{\prime}_{j}$, resulting in the set consisting of all elements $\bm{c}+\bm{c}^{\prime}$ for any $\bm{c}\in\mathcal{C}^{\prime}_{i}$ and $\bm{c}^{\prime}\in\mathcal{C}^{\prime}_{j}$, and where $(a)$ follows due to the fact that the Hadamard product is distributive over addition.

The symbols requested by the user are then obtained solving the system of linear equations defined by

For the retrieval, we require $\tilde{\mathcal{C}}$ to be a valid code, i.e., it must have a code rate strictly less than $1$. For a given number of colluding SBSs $T$, the combination of conditions on $\bar{\mathcal{C}}$ and $\tilde{\mathcal{C}}$ restricts the choice for the underlying storage codes $\{\mathcal{C}_{i}\}$. In the following theorem, we present a family of MDS codes, namely generalized Reed-Solomon (GRS) codes, that work with the protocol. A GRS code $\mathcal{C}$ over $\mathrm{GF}(q)$ of length $n$ and dimension $k$ is a weighted polynomial evaluation code of degree $k$ defined by some weighting vector $\bm{v}=(v_{1},\ldots,v_{n})\in(\mathrm{GF}(q)^{\times})^{n}$ and an evaluation vector $\bm{\kappa}=(\kappa_{1},\ldots,\kappa_{n})\in(\mathrm{GF}(q)^{\times})^{n}$ satisfying $\kappa_{i}\neq\kappa_{j}$ for all $i\neq j$ [25, Ch. 5]. In the sequel, we refer to $(n,k,\bm{v},\bm{\kappa})$ as the parameters of a GRS code $\mathcal{C}$.

Note that the retrieval code $\bar{\mathcal{C}}$ depends on the $n$ SBSs within visibility that are contacted by the user through its evaluation vector. Finally, we remark that, with some slight modifications, the proposed protocol can be adapted to work with non-MDS codes.

As an example, consider the case of $F=2$ files, $\bm{X}^{(1)}$ and $\bm{X}^{(2)}$, both of size $\beta L$ bits. The first file $\bm{X}^{(1)}$ is stored in the SBSs according to Fig. 2 using an $(N_{\mathsf{SBS}}=6,k_{1}=1)$ binary repetition code $\mathcal{C}_{1}$. Similarly, the second file $\bm{X}^{(2)}$ is stored (again according to Fig. 2) using an $(N_{\mathsf{SBS}}=6,k_{2}=5)$ binary single parity-check code $\mathcal{C}_{2}$. Assume $n=N_{\mathsf{SBS}}=6$ (i.e., no puncturing) and that none of the SBSs collude, i.e., $T=1$. Furthermore, we assume that the user wants to retrieve $\bm{X}^{(1)}$ and is able to contact $b=n=6$ SBSs (i.e., we consider the extreme case where the user is not contacting the MBS). According to Theorem 1, we can choose $\beta=\Gamma=n-(k_{\mathsf{max}}+T-1)=6-(5+1-1)=1$ and $d=k_{\mathsf{max}}=5$. Finally, we choose $\bar{\mathcal{C}}$ as an $(n=6,T=1)$ binary repetition code.

According to (7), the retrieval code $\tilde{\mathcal{C}}=(\mathcal{C}_{1}+\mathcal{C}_{2})\circ\bar{\mathcal{C}}=\mathcal{C}_{1}+\mathcal{C}_{2}=\mathcal{C}_{2}$ and can be generated by

Moreover, let

where $\mathcal{I}_{1}$ is an information set of $\mathcal{C}_{\mathsf{max}}=\mathcal{C}_{2}$ (the submatrix $\bm{G}^{\mathcal{C}_{2}}|_{\mathcal{I}_{1}}$ has rank $k_{2}=5$). Note that $\hat{\bm{E}}$ satisfies all three conditions $\mathsf{C1}$–$\mathsf{C3}$ and has column weight profile $(1,1,1,1,1,0)=(|\mathcal{F}_{1}|,\ldots,|\mathcal{F}_{6}|)$.

Query Construction. The user generates $\beta F=2$ codewords $\bar{\bm{c}}^{(1)}_{1}$ and $\bar{\bm{c}}^{(2)}_{1}$ independently and uniformly at random from $\bar{\mathcal{C}}$. Without loss of generality, let $\bar{\bm{c}}^{(1)}_{1}=\bar{\bm{c}}^{(2)}_{1}=(1,\ldots,1)$. Next, the $n=6$ subqueries $q_{1}^{(l)}$, $l=1,\ldots,6$, are constructed according to 4, 5 as

where $\mathring{\bm{c}}_{l}$ is defined in (3).

File Retrieval. Consider the $n=6$ subresponses $r_{1}^{(l)}$, $l=1,\ldots,6$. Then, according to (III-B),

and the code symbol $x_{1,1}^{(1)}$ of the file $\bm{X}^{(1)}$ is recovered from

Note that in order to retain privacy across the two files of the library, we need to send $d=k_{\mathsf{max}}=5$ subqueries to each SBS, thus generating $5$ subresponses from each SBS (even if the first file can be recovered from the $n=6$ subresponses $r_{1}^{(l)}$, $l=1,\ldots,6$).