Private Information Retrieval from Locally Repairable Databases with Colluding Servers

TL;DR

This paper introduces a novel private information retrieval scheme for locally repairable codes with colluding servers, achieving high rates and small field sizes, applicable to a broad class of storage codes.

Contribution

It presents the first PIR scheme for maximally recoverable LRCs based on linearized Reed--Solomon codes, with optimal field size and broad applicability.

Findings

Achieves the smallest field sizes among MR-LRCs for many parameters.

Attains higher PIR rates than existing schemes for any MDS code.

Compatible with network coding, maintaining high rates with polynomial field sizes.

Abstract

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups $ g $, locality $ r $, local distance $ \delta $ and dimension $ k $. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed--Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only $ r $ nodes (out of $ r+\delta-1 $) are downloaded per server. The PIR scheme achieves the (download) rate $ R = (N - k - rt + 1)/N $, where $ N = gr $ is the length of the MDS code obtained after removing the local parities, and for any $ t $ colluding servers such that $ k + rt \leq N $. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed--Solomon codes, but Reed--Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

Equations

$$R=\frac{N-k-\tau +1}{N}\cdot {\left(1-{\left(\frac{k+\tau -1}{N}\right)}^m\right)}^{-1},$$

$${\mathcal{D}}_a^i(\beta )={\sigma }^i(\beta )N_i(a),$$

$$x\beta =\sigma (\beta )x,$$

$$F^{{\mathcal{D}}_a}=\sum _{i=0}^d{F}_i{\mathcal{D}}_a^i\in {\mathbb{F}}_{q^r}[{\mathcal{D}}_a],$$

$$F^{{\mathcal{D}}_a}(\mathbf{\beta })=(F^{{\mathcal{D}}_a}({\beta }_1),F^{{\mathcal{D}}_a}({\beta }_2),\dots ,F^{{\mathcal{D}}_a}({\beta }_r))\in {\mathbb{F}}_{q^r}^r.$$

$$F^{{\mathcal{D}}_{\mathbf{a}}}(\mathbf{\beta })=(F^{{\mathcal{D}}_{a_1}}(\mathbf{\beta }),F^{{\mathcal{D}}_{a_2}}(\mathbf{\beta }),\dots ,F^{{\mathcal{D}}_{a_g}}(\mathbf{\beta }))\in {\mathbb{F}}_{q^r}^N,$$

$${\mathcal{C}}_{N,k}(\mathbf{a},\mathbf{\beta })=\{F^{{\mathcal{D}}_{\mathbf{a}}}(\mathbf{\beta })\in {\mathbb{F}}_{q^r}^N\mid F\in {\mathbb{F}}_{q^r}[x;\sigma ],\mathrm{deg}(F)<k\text{ or }F=0\}\subseteq {\mathbb{F}}_{q^r}^N.$$

$${\mathcal{C}}_{glob}={\mathcal{C}}_{out}{\mathrm{Diag}}_g(A)\subseteq {\mathbb{F}}_{q^r}^n,$$

$$\operatorname{Diag}(A,A,\ldots,A)=\left(\begin{array}[]{cccc}A&0&\ldots&0\\ 0&A&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&A\end{array}\right).$$

$$X=\left(\begin{array}[]{c}\mathbf{x}^{1}\\ \vdots\\ \mathbf{x}^{m}\end{array}\right)\in\mathbb{F}_{q^{r}}^{bm\times k}.$$

$$G_{glob}=G_{out}{\mathrm{Diag}}_g(A)\in {\mathbb{F}}_{q^r}^{k\times n}$$

$$Z=XG_{out}=\left(\begin{array}[]{c}\mathbf{z}^{1}\\ \vdots\\ \mathbf{z}^{m}\end{array}\right)=\left(\mathbf{z}_{1},\mathbf{z}_{2},\ldots,\mathbf{z}_{g}\right)=\left(\begin{array}[]{ccc}\mathbf{z}_{1}^{1}&\ldots&\mathbf{z}_{g}^{1}\\ \vdots&\ddots&\vdots\\ \mathbf{z}_{1}^{m}&\ldots&\mathbf{z}_{g}^{m}\end{array}\right)\in\mathbb{F}_{q^{r}}^{bm\times N}.$$

$${\mathbf{q}}_j^i=({\mathbf{q}}_j^{i,1},{\mathbf{q}}_j^{i,2},\dots ,{\mathbf{q}}_j^{i,m})\in {\mathbb{F}}_{q^r}^{bmr},$$

$$R=\frac{bk}{Ns}.$$

$$I(({\mathbf{q}}_j^i{)}_{j\in T};i)=0,$$

$$M_{\boldsymbol{\beta}}\left(\mathbf{x}\right)=\left(\begin{array}[]{cccc}x^{1}_{1}&x^{1}_{2}&\ldots&x^{1}_{r}\\ x^{2}_{1}&x^{2}_{2}&\ldots&x^{2}_{r}\\ \vdots&\vdots&\ddots&\vdots\\ x^{r}_{1}&x^{r}_{2}&\ldots&x^{r}_{r}\\ \end{array}\right),$$

$$\mathbf{x}\star \mathbf{y}=M_{\mathbf{\beta }}^{-1}(M_{\mathbf{\beta }}(\mathbf{x})M_{\mathbf{\beta }}(\mathbf{y}))\in {\mathbb{F}}_{q^r}^r.$$

$$\mathbf{x}\ast \mathbf{y}=({\mathbf{x}}_1\star {\mathbf{y}}_1,{\mathbf{x}}_2\star {\mathbf{y}}_2,\dots ,{\mathbf{x}}_g\star {\mathbf{y}}_g)\in {\mathbb{F}}_{q^r}^N,$$

$$\mathbf{x}\cdot \mathbf{y}=\sum _{j=1}^g{\mathbf{x}}_j\star {\mathbf{y}}_j\in {\mathbb{F}}_{q^r}^r.$$

$$M_{\mathbf{\beta }}^{-1}(M_{\mathbf{\beta }}(\mathbf{x})M_{\mathbf{\beta }}(\mathbf{y}))=\sum _{i=1}^r{x}_i{\mathbf{y}}^i\in {\mathbb{F}}_{q^r}^r.$$

$$(FG{)}^{{\mathcal{D}}_a}=F^{{\mathcal{D}}_a}\circ G^{{\mathcal{D}}_a}.$$

$${\mathcal{D}}_a\circ (\beta \mathrm{Id})=\sigma (\beta ){\mathcal{D}}_a,$$

$$(FG{)}^{{\mathcal{D}}_{\mathbf{a}}}(\mathbf{\beta })=F^{{\mathcal{D}}_{\mathbf{a}}}(\mathbf{\beta })\ast G^{{\mathcal{D}}_{\mathbf{a}}}(\mathbf{\beta }),$$

$${\sigma }^{\ell }(\mathbf{\beta })\star \mathbf{y}={\sigma }^{\ell }(\mathbf{y}),$$

$${\sigma }^{\ell }(\mathbf{\beta })\star \mathbf{y}=\sum _{i=1}^r{\sigma }^{\ell }({\beta }_i){\mathbf{y}}^i={\sigma }^{\ell }\left(\sum _{i=1}^r{\beta }_i{\mathbf{y}}^i\right)={\sigma }^{\ell }(\mathbf{y}),$$

$${\mathcal{D}}_a^{\ell }(\mathbf{\beta })\star \mathbf{y}=(N_{\ell }(a){\sigma }^{\ell }(\mathbf{\beta }))\star \mathbf{y}=N_{\ell }(a)({\sigma }^{\ell }(\mathbf{\beta })\star \mathbf{y})=N_{\ell }(a){\sigma }^{\ell }(\mathbf{y})={\mathcal{D}}_a^{\ell }(\mathbf{y}),$$

$${\mathcal{C}}_1\ast {\mathcal{C}}_2=⟨\{{\mathbf{c}}_1\ast {\mathbf{c}}_2\mid {\mathbf{c}}_1\in {\mathcal{C}}_1,{\mathbf{c}}_2\in {\mathcal{C}}_2\}⟩\subseteq {\mathbb{F}}_{q^r}^N,$$

$${\mathcal{C}}_{N,k_1}(\mathbf{a},\mathbf{\beta })\ast {\mathcal{C}}_{N,k_2}(\mathbf{a},\mathbf{\beta })={\mathcal{C}}_{N,k_1+k_2-1}(\mathbf{a},\mathbf{\beta }),$$

$$\mathrm{deg}(FG)=\mathrm{deg}(F)+\mathrm{deg}(G),$$

$$k+rt\le N,$$

Full text

Keywords: Distributed storage, information-theoretical privacy, linearized Reed-Solomon codes, locally repairable codes, maximally recoverable codes, private information retrieval.

1 Introduction

Private information retrieval (PIR), introduced in [6, 7], consists in retrieving a file from a database without revealing the index of the retrieved file to the servers. In this work, we consider information-theoretical privacy. Originally [6, 7], databases were considered to store files using a repetition code, that is, each server stores one copy of each file, and servers were not considered to communicate with each other (collude) in order to gain information on the file index.

As it was pointed out in these seminal works, an obvious solution to the PIR problem is to download the entire database. However, this turns out to be wasteful and much higher download rates, or simply rates (the size of the file divided by the amount of downloaded data), can be achieved when more than one server is used. It was also shown in these seminal works that downloading the whole database is the only solution in the single-server case (for information-theoretical privacy). PIR from a database where data is stored after encoded by a non-repetition code (coded database) was considered in [2, 10, 29]. However, the number of servers in these works is either larger than the number of files or grows as the overhead of the storage code decreases, which are not practical scenarios.

Explicit PIR schemes from a database that uses an MDS storage code, and for $\tau\geq 1$ colluding servers, were obtained in [11, 34, 35, 39]. The optimal rate for a fixed number $m$ of stored files is

[TABLE]

for the cases $\tau=1$ [1, 30] and $k=1$ [31], although (1) is not optimal in general [32]. If the MDS storage code has dimension $k$ and length $N$, universal PIR schemes (compatible with any $(N,k)$ MDS storage code) were obtained in [34, 35] with rate $R=1/N$, for $\tau=N-k$, and rate $R=(N-k)/N$ for $\tau=1$. For $\tau=1$ or $k=1$, these rates become optimal exponentially fast as the number of files increases, according to (1). For $\tau>1$ and $k>1$, the optimal rate is unknown in general, but the rate of the PIR scheme in [11], $R=(N-k-\tau+1)/N$, is the highest known for an unrestricted number of files and is always strictly larger than that of known universal PIR schemes [34]. Unfortunately, the scheme in [11] works for (generalized) Reed-Solomon storage codes [27], but not for other MDS storage codes. Recently, it has been shown in [9] that the rate in [11] can be improved for a finite number of files $m$ to the rate in (1). However, the improved rate (1) reduces again to $(N-k-\tau+1)/N$ exponentially fast as the number of files increases.

As pointed out in the distributed storage literature however, MDS codes are not suitable for large databases, since repairing a failed node requires contacting and downloading the content of a large number of nodes, resulting in high repair latency. Locally repairable codes (LRCs), introduced in [14, 17] and already applied in practice [15, 28], allow to repair a single erasure (or generally $\delta-1$ erasures per local group, for a local distance $\delta$) by contacting at most a number $r$, called locality, of other nodes. At the same time, they may correct $d-1$ global erasures in catastrophic cases, where $d$ is the minimum distance of the LRC. Maximally recoverable (MR) LRCs were introduced in [3, 13] and allow to correct any information-theoretically correctable erasure pattern for given parameters $k$, $r$, $\delta$ and number of local groups $g$. MR-LRCs have optimal distance, but in general LRCs with optimal distance (such as those in [37]) are not MR.

Although some works in PIR consider general storage codes [11, 18], no work has achieved rates comparable to [11] for optimal LRC or MR-LRC databases, to the best of our knowledge.

In this work, we provide a PIR scheme for the class of MR-LRC storage codes introduced in [22], based on linearized Reed-Solomon codes [20]. With parameters as above, the total length of the MR-LRC code is $n=g(r+\delta-1)$ and the required field size is roughly $g^{r}$, which is the smallest known for MR-LRCs in the paremeter regime $r\leq gr-k$ (see [22]).

Since nodes inside a local group need to communicate much more frequently than with other nodes, we identify bijectively servers with local groups. Since there are $\delta-1$ redundant nodes per local group, we only consider and download information from the $r$ remaining nodes. In this way, we may think of $t$ colluding servers as $\tau=rt$ colluding nodes (although not any $rt$ colluding nodes correspond to some $t$ colluding servers). The remaining storage code after removing the local redundancies will be an MDS linearized Reed-Solomon code [20] of length $N=gr=n-g(\delta-1)$ and dimension $k$. Our PIR scheme achieves the rate $R=(N-k-rt+1)/N$, coinciding with the highest known for MDS storage codes for an unrestricted number of stored files [11].

On the mathematical side, to provide our PIR scheme, we introduce inner and coordinate-wise matrix products, which simultaneously extend matrix products and classical coordinate-wise products (also known as Schur products). We show in Corollary 11 that the coordinate-wise matrix product of two linearized Reed-Solomon codes is again a linearized Reed-Solomon code, which recovers as particular cases the corresponding results on the matrix product of Gabidulin codes [23, Lemma 10] (see also [36]) and the Schur product of Reed-Solomon codes [11, Prop. 3]. Finally, it is worth mentioning that the storage codes and PIR scheme in [11] are recovered from ours simply by setting $r=\delta=1$ throughout this paper.

We remark that a PIR scheme from Gabidulin storage codes [12] has been recently given in [36]. Gabidulin codes can be used to construct MR-LRCs similarly to linearized Reed-Solomon codes (see [4, 26]), and the PIR scheme in [36] can be used for such MR-LRCs in the same way as ours. However, the required field size would be at least $2^{gr}$ (see Subsection 6.3), which is exponential in the number of servers $g$ and the code length $N=gr$, in contrast with polynomial field sizes $g^{r}$ for our scheme (note also that $r$ is preferably small). The main objective in [36] is to give a PIR scheme where communication between the user and each server is through a linearly coded network [19]. We will see in Subsection 6.3 that our PIR scheme can be used in the same scenario, achieving the same rate, but with polynomial field sizes as noted above.

The remainder of this paper is organized as follows. In Section 2, we collect preliminaries on linearized Reed-Solomon codes and MR-LRCs. In Section 3, we formulate general PIR schemes for MR-LRC databases. In Section 4, we develop the mathematical tools regarding inner and coordinate-wise matrix products for our PIR scheme. In Section 5, we give our particular PIR scheme. Finally, in Section 6, we discuss some further considerations.

2 Preliminaries

Notation

Let $q$ be a prime power. We will denote by $\mathbb{F}$ an arbitrary field, and by $\mathbb{F}_{q}$ the finite field with $q$ elements. Usually, we will consider $\mathbb{F}=\mathbb{F}_{q^{r}}$, where $r\geq 1$ will be the locality of the considered storage codes.

We will also denote by $\mathbb{F}^{m\times n}$ the set of $m\times n$ matrices with entries in $\mathbb{F}$, and we denote $\mathbb{F}^{n}=\mathbb{F}^{1\times n}$. For a positive integer $n$, we denote $[n]=\{1,2,\ldots,n\}$. Given $\mathcal{R}\subseteq[n]$, we denote by $\mathbf{c}_{\mathcal{R}}\in\mathbb{F}^{|\mathcal{R}|}$, $A|_{\mathcal{R}}\in\mathbb{F}^{m\times|\mathcal{R}|}$ and $\mathcal{C}_{\mathcal{R}}\subseteq\mathbb{F}^{|\mathcal{R}|}$ the restrictions of a vector $\mathbf{c}\in\mathbb{F}^{n}$, a matrix $A\in\mathbb{F}^{m\times n}$ and a code $\mathcal{C}\subseteq\mathbb{F}^{n}$, respectively, to the coordinates indexed by $\mathcal{R}$.

2.1 Linearized Reed-Solomon Codes

In this subsection, we recall the definition of linearized Reed-Solomon codes, which were introduced in [20] and are natural hybrids between Reed-Solomon codes [27] and Gabidulin codes [12]. They are the main building blocks of the maximally recoverable and locally repairable storage codes that we consider in this work (see the next subsection), which were introduced in [22].

Fix $r\geq 1$ and let $\sigma:\mathbb{F}_{q^{r}}\longrightarrow\mathbb{F}_{q^{r}}$ be the Fröbenius field automorphism, given by $\sigma(a)=a^{q}$, for all $a\in\mathbb{F}_{q^{r}}$. We next define linear operators as in [20, Def. 20].

Definition 1** ([20]).**

Fix $a\in\mathbb{F}_{q^{r}}$, and define its $i$th norm as $N_{i}(a)=\sigma^{i-1}(a)\cdots\sigma(a)a$, for $i\in\mathbb{N}$. We define the $\mathbb{F}_{q}$-linear operator $\mathcal{D}_{a}^{i}:\mathbb{F}_{q^{r}}\longrightarrow\mathbb{F}_{q^{r}}$ by

[TABLE]

for all $\beta\in\mathbb{F}_{q^{r}}$, and all $i\in\mathbb{N}$. Define also $\mathcal{D}_{a}=\mathcal{D}_{a}^{1}$ and observe that $\mathcal{D}_{a}^{i+1}=\mathcal{D}_{a}\circ\mathcal{D}_{a}^{i}$, for $i\in\mathbb{N}$. Denote by $\mathbb{F}_{q^{r}}[\mathcal{D}_{a}]$ the polynomial ring in the operator $\mathcal{D}_{a}$, for $a\in\mathbb{F}_{q^{r}}$.

Recall that the skew polynomial ring $\mathbb{F}_{q^{r}}[x;\sigma]$, introduced in [25], is the polynomial ring on the variable $x$ but with non-commutative product given by the rule

[TABLE]

for all $\beta\in\mathbb{F}_{q^{r}}$. For $F=\sum_{i=0}^{d}F_{i}x^{i}\in\mathbb{F}_{q^{r}}[x;\sigma]$, we define

[TABLE]

for $a\in\mathbb{F}_{q^{r}}$. In the following, for $F=\sum_{i=0}^{d}F_{i}x^{i}\in\mathbb{F}_{q^{r}}[x;\sigma]$, for $a\in\mathbb{F}_{q^{r}}$ and for $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{r})\in\mathbb{F}_{q^{r}}^{r}$, where $\beta_{i}\in\mathbb{F}_{q^{r}}$, for $i=1,2,\ldots,r$, we use the notation

[TABLE]

Next, given $\mathbf{a}=(a_{1},a_{2},\ldots,a_{g})\in\mathbb{F}_{q^{r}}^{g}$, we define the total evaluation vector of $F$ at $(\mathbf{a},\boldsymbol{\beta})$ as

[TABLE]

where $N=gr$. The following definition is a particular case of [20, Def. 31]. See also [22, Def. 3].

Definition 2** (Linearized Reed-Solomon codes [20]).**

Fix a primitive element $\gamma\in\mathbb{F}_{q^{r}}^{*}$ and let $\mathbf{a}=(\gamma^{0},\gamma^{1},\ldots,\gamma^{g-1})$ $\in\mathbb{F}_{q^{r}}^{g}$. Fix an ordered basis $\boldsymbol{\beta}=(\beta_{1},$ $\beta_{2},$ $\ldots,\beta_{r})$ $\in\mathbb{F}_{q^{r}}^{r}$ of $\mathbb{F}_{q^{r}}$ over $\mathbb{F}_{q}$. For $k=0,1,2,\ldots,N=gr$, we define the $(N,k)$ linearized Reed-Solomon (LRS) code as

[TABLE]

Here, the degree $\deg(F)$ of a non-zero skew polynomial $F=\sum_{i\in\mathbb{N}}F_{i}x^{i}\in\mathbb{F}_{q^{r}}[x;\sigma]$, where $F_{i}\in\mathbb{F}_{q^{r}}$ for $i\in\mathbb{N}$, is defined as the maximum $i\in\mathbb{N}$ such that $F_{i}\neq 0$.

Linearized Reed-Solomon codes recover Reed-Solomon codes [27] by setting $r=1$ and $\beta_{1}=1$, and they recover Gabidulin codes [12] by setting $g=1$. See also [21, Table II].

2.2 Locally Repairable Codes with Maximal Recoverability

In this subsection, we briefly recall the definitions of locally repairable codes [14, 17] and maximal recoverability [3, 13], and we revisit the construction based on linearized Reed-Solomon codes from [22].

Consider $\mathbb{F}$ to be an arbitrary finite field. As in [3, 13], we will consider disjoint local groups.

Definition 3** (Locally repairable codes [14, 17]).**

We say that a code $\mathcal{C}\subseteq\mathbb{F}^{n}$ is a locally repairable code (LRC) with $(r,\delta)$ localities if there exists a partition $[n]=\Gamma_{1}\cup\Gamma_{2}\cup\ldots\cup\Gamma_{g}$, where $\Gamma_{i}\cap\Gamma_{j}=\varnothing$ if $i\neq j$, such that

$|\Gamma_{j}|=r+\delta-1$, and 2. 2.

${\rm d}_{H}(\mathcal{C}_{\Gamma_{j}})\geq\delta$,

for $j=1,2,\ldots,g$. The set $\Gamma_{j}$ is called the $j$th local group, $r$ is called the locality, and $\delta$ is called the local distance.

Note that $n=(r+\delta-1)g$, but necessarily it must hold that $k=\dim(\mathcal{C})\leq gr$. Usually, the restricted code $\mathcal{C}_{\Gamma_{j}}$ is a subcode of an $(r+\delta-1,r)$ MDS local linear code that can repair any $\delta-1$ erasures in $\Gamma_{j}$ from the other $r$ nodes in $\Gamma_{j}$. Maximally recoverable LRCs, introduced in [3, Def. 2.1] and [13, Def. 6], can correct any $h$ more erasures than the Cartesian product of the MDS local codes (the case $h=0$), where $h=gr-k\geq 0$ is the number of global parities. These are all information-theoretically correctable erasure patterns for given $r$, $\delta$, $k$ and $g$. Formally:

Definition 4** (Maximal recoverability [3, 13]).**

We say that an LRC $\mathcal{C}\subseteq\mathbb{F}^{n}$ with $(r,\delta)$ localities is maximally recoverable (MR) if, for any $\Delta_{j}\subseteq\Gamma_{j}$ with $|\Delta_{j}|=r$, for $j=1,2,\ldots,g$, the restricted code $\mathcal{C}_{\Delta}\subseteq\mathbb{F}^{N}$ is MDS, where $\Delta=\bigcup_{j=1}^{g}\Delta_{j}$ and $N=|\Delta|=gr$. We say for short that $\mathcal{C}$ is MR-LRC. We will usually call $\mathcal{C}_{\Delta}\subseteq\mathbb{F}^{N}$ a remaining MDS code of $\mathcal{C}$ (there is one for each choice of $\Delta_{j}$’s).

In general, MR-LRCs can correct more erasure patterns than LRCs with optimal minimum distance with respect to the bound in [17, Th. 2.1], such as Tamo-Barg codes [37].

In [22, Const. 1], a construction of MR-LRCs was given based on linearized Reed-Solomon codes (Definition 2). This construction recovers Reed-Solomon codes with local replication when $r=1$, and it recovers Cartesian products of local codes when $k=N=gr$ (that is, $h=0$).

Construction 1** (LRS-based MR-LRC [22]).**

Fix the positive integers $g$, $r$ and $\delta$, and choose any base field size $q>g$. Next choose a dimension $k=1,2,\ldots,N=gr$, and:

Outer code: An $(N,k)$ linearized Reed-Solomon code $\mathcal{C}_{out}=\mathcal{C}_{N,k}(\mathbf{a},\boldsymbol{\beta})\subseteq\mathbb{F}_{q^{r}}^{N}$ as in Definition 2. 2. 2.

Local codes: Any linear $(r+\delta-1,r)$ MDS code $\mathcal{C}_{loc}\subseteq\mathbb{F}_{q}^{r+\delta-1}$. 3. 3.

Global code: Let $\mathcal{C}_{glob}\subseteq\mathbb{F}_{q^{r}}^{n}$, with $n=(r+\delta-1)g=N+(\delta-1)g$, be given by

[TABLE]

where $A\in\mathbb{F}_{q}^{r\times(r+\delta-1)}$ is any generator matrix of $\mathcal{C}_{loc}$ and $\operatorname{Diag}_{g}(A)=\operatorname{Diag}(A,A,\ldots,A)\in\mathbb{F}_{q}^{N\times n}$ is the block-diagonal matrix with $A$ repeated in the main block-diagonal $g$ times:

[TABLE]

Note that we also need that $q\geq r+\delta-1$ in Construction 1, but we are assuming that $q>g$ and in general it is desirable that $g\gg r+\delta-1$, hence we do not stress that $q\geq r+\delta-1$.

The following result is [22, Th. 2] and states the MR and LRC properties of the global code $\mathcal{C}_{glob}$ in Construction 1.

Theorem 1** ([22]).**

Let $\mathcal{C}_{glob}\subseteq\mathbb{F}_{q^{r}}^{n}$ be the global code from Construction 1, and let $\Gamma_{j}\subseteq[n]$ be the subset of coordinates ranging from $(r+\delta-1)(j-1)+1$ to $(r+\delta-1)j$, for $j=1,2,\ldots,g$. Then the code $\mathcal{C}_{glob}\subseteq\mathbb{F}_{q^{r}}^{n}$ has $(r,\delta)$ localities, local groups $\Gamma_{1},\Gamma_{2},\ldots,\Gamma_{g}$, and is maximally recoverable.

Observe that the field size required by $\mathcal{C}_{glob}$ in Construction 1 is $|\mathbb{F}|\approx g^{r}$. When $r\leq h=gr-k$, these codes are the MR-LRCs with smallest field sizes known so far. If $r$ is small and constant, global decoding with $\mathcal{C}_{glob}$ has comparable complexity to that of Tamo-Barg codes or Reed-Solomon codes (see [22, Ex. 2] and [22, Subsec. VI-B]).

3 Private Information Retrieval from MR-LRC Databases

In this section, we describe the private information retrieval (PIR) model that we consider in this work. Since we focus on MR-LRC databases, we make the following two observations:

Since communication is much more frequent and necessary among nodes inside a local group, we consider local groups $\Gamma_{j}\subseteq[n]$, rather than individual nodes $j\in[n]$, as corruptable units. In other words, the $j$th server will be identified with the $j$th local group, storing $r+\delta-1$ symbols from each codeword of the LRC, one symbol per node. Thus a subset $T\subseteq[g]$ of colluding servers is the same as the corresponding $(r+\delta-1)|T|$ colluding nodes. 2. 2.

To help reduce the downloaded amount of data from the $j$th server (i.e. $j$th local group, see Item 1), we assume that only $r$ stored symbols from each codeword are downloaded from that server, since the remaining $\delta-1$ nodes only contain locally redundant information.

In other words, we consider private information retrieval from the remaining MDS coded data obtained by removing all local redundancies (Definition 4), where colluding servers are identified with colluding local groups of nodes. Since the remaining MDS code in Construction 1 is a linearized Reed-Solomon code (Definition 2), the main objective of the rest of the paper is to obtain a tailor-made PIR scheme for linearized Reed-Solomon codes, where our strategy will be to protect against any $\tau=rt$ colluding nodes. Note however that, after removing the local redundancies, not all sets of $rt$ colluding nodes correspond to some $t$ colluding servers.

Fix positive integers $b$, $m$ and $k\leq N=gr$, and let $\mathbf{x}^{1},$ $\mathbf{x}^{2},$ $\ldots,$ $\mathbf{x}^{m}\in\mathbb{F}_{q^{r}}^{b\times k}$ be the $m$ files to be stored. Arrange them as

[TABLE]

Let $G_{out}\in\mathbb{F}_{q^{r}}^{k\times N}$ be a generator matrix of the $k$-dimensional outer code $\mathcal{C}_{out}\subseteq\mathbb{F}_{q^{r}}^{N}$ in Construction 1. Then

[TABLE]

is the generator matrix of the global code $\mathcal{C}_{glob}\subseteq\mathbb{F}_{q^{r}}^{n}$ in Construction 1, where $A\in\mathbb{F}_{q}^{r\times(r+\delta-1)}$ is a generator matrix of an $(r+\delta-1,r)$ MDS code. Recall that $N=gr$ and $n=g(r+\delta-1)$.

Each file $\mathbf{x}^{i}$ is then encoded into $\mathbf{y}^{i}=\mathbf{x}^{i}G_{glob}\in\mathbb{F}_{q^{r}}^{b\times n}$, where $\mathbf{y}^{i}_{\Gamma_{j}}\in\mathbb{F}_{q^{r}}^{b\times(r+\delta-1)}$ is stored in the $j$th server, and where $\Gamma_{j}$ is as in Theorem 1. Let $\Delta_{j}\subseteq\Gamma_{j}$ be the first $r$ coordinates in $\Gamma_{j}$ and assume that the first $r$ columns of $A$ form the identity matrix (i.e. $A$ is systematic). Then if we disregard the nodes in $\Gamma_{j}\setminus\Delta_{j}$, the part of the $i$th file stored in the $j$th server is $\mathbf{z}^{i}_{j}\in\mathbb{F}_{q^{r}}^{b\times r}$, for $j=1,2,\ldots,g$, where

[TABLE]

Thus the remaining MDS code coincides with the outer code in Construction 1, $\mathcal{C}_{\Delta}=\mathcal{C}_{out}$, which is a linearized Reed-Solomon code (Definition 2). We may now formalize a general private information retrieval scheme for such a code.

Definition 5**.**

A private information retrieval (PIR) scheme for the MR-LRC distributed storage system described above consists, for each $i=1,2,\ldots,m$, of:

Random queries

[TABLE]

sent to the $j$th server to retrieve the $i$th file, where $\mathbf{q}_{j}^{i,\ell}\in\mathbb{F}_{q^{r}}^{br}$, for $\ell=1,2,\ldots,m$ and $j=1,2,\ldots,g$. 2. 2.

The corresponding response $\mathbf{r}_{j}^{i}=\mathbf{z}_{j}\cdot\mathbf{q}_{j}^{i}\in\mathbb{F}_{q^{r}}^{r}$ of the $j$th server when requested the $i$th file (note that the server only knows $\mathbf{q}_{j}^{i}$ in principle), for $j=1,2,\ldots,g$, where $\mathbf{z}_{j}\in\mathbb{F}_{q^{r}}^{bmN}$ and the product $\cdot$ will be given in (13). We denote $\mathbf{r}^{i}=(\mathbf{r}_{1}^{i},\mathbf{r}_{2}^{i},\ldots,\mathbf{r}_{g}^{i})\in\mathbb{F}_{q^{r}}^{N}$. 3. 3.

A number $s$ of iterations of Items 1 and 2, until the $i$th file can be recovered from the responses $\mathbf{r}^{i}$ in Item 2. 4. 4.

A reconstruction function with input the $s$ responses $\mathbf{r}^{i}$ and output the $i$th file $\mathbf{x}^{i}$.

Note that one major difference with [11, Def. 4] and the one-shot schemes in [9] is that we do not use the usual inner product $\mathbf{z}\cdot\mathbf{q}$, but a generalization of it (see Definition 8).

As usual in the PIR literature, our goal is to maximize the download rate, which is defined as the file size divided by the amount of downloaded data. The upload cost may be considered negligible by further folding the scheme $b^{\prime}\gg 1$ times, thus a total of $bb^{\prime}$ times. In our case, we disregard local redundancies, hence the download rate is as follows.

Definition 6**.**

We define the download rate, or simply rate, of a PIR scheme given as in Definition 5 as

[TABLE]

We require information-theoretical privacy for a given number $t$ of colluding servers (i.e. colluding local groups).

Definition 7**.**

We say that a PIR scheme as in Definition 5 protects against $t$ colluding servers if, for every $T\subseteq[g]$ of size $t$, in each iteration of the scheme we have that

[TABLE]

where $I(X;Y)$ denotes the mutual information between two random variables $X$ and $Y$ (see [8, Ch. 12]).

4 Coordinate-wise and Inner Matrix products

In this section, we define and collect the main properties of inner and coordinate-wise matrix products, which will be used in our PIR scheme, as shown in Item 2 in Definition 5.

4.1 Definition and Linearity Properties

Fix an ordered basis $\boldsymbol{\beta}=(\beta_{1},$ $\beta_{2},$ $\ldots,$ $\beta_{r})\in\mathbb{F}_{q^{r}}^{r}$ of $\mathbb{F}_{q^{r}}$ over $\mathbb{F}_{q}$. Denote by $M_{\boldsymbol{\beta}}:\mathbb{F}_{q^{r}}^{r}\longrightarrow\mathbb{F}_{q}^{r\times r}$ the corresponding matrix-representation map, given by

[TABLE]

for $\mathbf{x}=(x_{1},x_{2},\ldots,x_{r})\in\mathbb{F}_{q^{r}}^{r}$, where $x_{j}^{1},x_{j}^{2},\ldots,x_{j}^{r}\in\mathbb{F}_{q}$ are the unique scalars such that $x_{j}=\sum_{i=1}^{m}\beta_{i}x_{j}^{i}\in\mathbb{F}_{q^{r}}$, for $j=1,2,\ldots,r$. Observe that $M_{\boldsymbol{\beta}}$ is an $\mathbb{F}_{q}$-linear vector space isomorphism, and it is the identity map if $r=1$ and $\beta_{1}=1$.

Definition 8**.**

Given $\mathbf{x},\mathbf{y}\in\mathbb{F}_{q^{r}}^{r}$, we define their matrix product with respect to the basis $\boldsymbol{\beta}$ as

[TABLE]

For $N=gr$, $\mathbf{x}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{g})\in\mathbb{F}_{q^{r}}^{N}$ and $\mathbf{y}=(\mathbf{y}_{1},$ $\mathbf{y}_{2},$ $\ldots,$ $\mathbf{y}_{g})\in\mathbb{F}_{q^{r}}^{N}$, where $\mathbf{x}_{j},\mathbf{y}_{j}\in\mathbb{F}_{q^{r}}^{r}$, for $j=1,2,\ldots,g$, we define their coordinate-wise matrix product as

[TABLE]

and we define their inner matrix product $\cdot$ as

[TABLE]

The products $\star$, $*$ and $\cdot$ all depend on the subfield $\mathbb{F}_{q}\subseteq\mathbb{F}_{q^{r}}$ (thus $q$ and $r$) and the ordered basis $\boldsymbol{\beta}$, but we will not denote this dependence for simplicity. The classical coordinate-wise and inner products in $\mathbb{F}_{q}^{N}$, used in [11] for PIR (and in general in the literature, see [9]), are recovered by setting $r=1$ and $\beta_{1}=1$ (thus $N=g$).

From the definitions, we note also that, if $\mathbf{x}=(x_{1},$ $x_{2},$ $\ldots,$ $x_{r})\in\mathbb{F}_{q^{r}}^{r}$ and $\mathbf{y}=\sum_{i=1}^{r}\beta_{i}\mathbf{y}^{i}\in\mathbb{F}_{q^{r}}^{r}$, with $x_{i}\in\mathbb{F}_{q^{r}}$ and $\mathbf{y}^{i}\in\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,r$, then

[TABLE]

From Equation (14) applied coordinate-wise, we deduce the following bilinearity properties.

Lemma 9**.**

The coordinate-wise matrix product $*$ is $\mathbb{F}_{q}$-bilinear and $\mathbb{F}_{q^{r}}$-linear in the first component, that is,

$(\mathbf{x}+\mathbf{x}^{\prime})*\mathbf{y}=\mathbf{x}*\mathbf{y}+\mathbf{x}^{\prime}*\mathbf{y}$* and $\mathbf{x}*(\mathbf{y}+\mathbf{y}^{\prime})=\mathbf{x}*\mathbf{y}+\mathbf{x}*\mathbf{y}^{\prime}$,* 2. 2.

$(a\mathbf{x})*\mathbf{y}=a(\mathbf{x}*\mathbf{y})$* and $\mathbf{x}*(b\mathbf{y})=b(\mathbf{x}*\mathbf{y})$,*

for all $\mathbf{x},\mathbf{x}^{\prime},\mathbf{y},\mathbf{y}^{\prime}\in\mathbb{F}_{q^{r}}^{N}$, all $a\in\mathbb{F}_{q^{r}}$ and all $b\in\mathbb{F}_{q}$.

4.2 Products of Skew and Linearized Polynomials

We have the following important connection between the rings $\mathbb{F}_{q^{r}}[x;\sigma]$ and $\mathbb{F}_{q^{r}}[\mathcal{D}_{a}]$, for all $a\in\mathbb{F}_{q^{r}}$. We consider $\mathbb{F}_{q^{r}}[\mathcal{D}_{a}]$ as a ring with conventional addition and with composition of maps as multiplication, denoted by $\circ$.

Lemma 10**.**

For all $F,G\in\mathbb{F}_{q^{r}}[x;\sigma]$ and all $a\in\mathbb{F}_{q^{r}}$, it holds that

[TABLE]

In particular, the map $\mathbb{F}_{q^{r}}[x;\sigma]\longrightarrow\mathbb{F}_{q^{r}}[\mathcal{D}_{a}]$ given by (4) is a (surjective) ring morphism.

Proof.

Observe that

[TABLE]

for all $\beta\in\mathbb{F}_{q^{r}}$, where ${\rm Id}=\mathcal{D}_{a}^{0}$ is the multiplicative identity of $\mathbb{F}_{q^{r}}[\mathcal{D}_{a}]$, and note that (15) coincides with (3) if we set $x=\mathcal{D}_{a}$. Since (3) is the defining property of the product in the skew polynomial ring $\mathbb{F}_{q^{r}}[x;\sigma]$, the result follows. ∎

The main result of this section is showing that products of skew polynomials become coordinate-wise matrix products after evaluation via the operators $\mathcal{D}_{a}$.

Theorem 2**.**

Let $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{r})\in\mathbb{F}_{q^{r}}^{r}$ be an ordered basis of $\mathbb{F}_{q^{r}}$ over $\mathbb{F}_{q}$, and let coordinate-wise matrix products $*$ be defined via $\boldsymbol{\beta}$. Then it holds that

[TABLE]

for all vectors $\mathbf{a}=(a_{1},a_{2},\ldots,a_{g})\in\mathbb{F}_{q^{r}}^{g}$ and all skew polynomials $F,G\in\mathbb{F}_{q^{r}}[x;\sigma]$.

Proof.

In [23, Prop. 1], it was proven that

[TABLE]

for all $\ell\in\mathbb{N}$ and all $\mathbf{y}\in\mathbb{F}_{q^{r}}^{r}$. We recall the proof of (16) for convenience of the reader. If $\mathbf{y}=\sum_{i=1}^{r}\beta_{i}\mathbf{y}^{i}$, where $\mathbf{y}^{i}\in\mathbb{F}_{q}^{r}$, for $i=1,2,\ldots,r$, then

[TABLE]

where the first equality is (14).

Since $\star$ is $\mathbb{F}_{q^{r}}$-linear in the first component (Lemma 9), and $\mathcal{D}_{a}^{\ell}=N_{\ell}(a)\sigma^{\ell}$, where $N_{\ell}(a)\in\mathbb{F}_{q^{r}}$, then we deduce from (16) that

[TABLE]

for all $a\in\mathbb{F}_{q^{r}}$, all $\mathbf{y}\in\mathbb{F}_{q^{r}}^{r}$ and all $\ell\in\mathbb{N}$. The case $g=1$ thus follows by combining the linearity on the first component of the matrix product (Lemma 9) with Lemma 10 for $g=1$.

Finally, the theorem for general $g$ follows by applying the case $g=1$ separately in each of the $g$ coordinates over the alphabet $\mathbb{F}_{q^{r}}^{r}$, and applying Lemma 10. ∎

Setting $r=1$ and $\beta_{1}=1$, the previous theorem is nothing but the well-known fact that coordinate-wise evaluation transforms conventional polynomial products into the conventional coordinate-wise product.

We conclude by deducing that the product of two linearized Reed-Solomon codes over the same ordered basis $\boldsymbol{\beta}$ is again a linearized Reed-Solomon code. For this purpose, given $\mathbb{F}_{q^{r}}$-linear codes $\mathcal{C}_{1},\mathcal{C}_{2}\subseteq\mathbb{F}_{q^{r}}^{N}$, we define their coordinate-wise matrix product as

[TABLE]

where $\langle\mathcal{A}\rangle$ may denote either the $\mathbb{F}_{q}$-linear or the $\mathbb{F}_{q^{r}}$-linear vector space generated by $\mathcal{A}\subseteq\mathbb{F}_{q^{r}}^{N}$.

Corollary 11**.**

Let $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{r})\in\mathbb{F}_{q^{r}}^{r}$ be an ordered basis of $\mathbb{F}_{q^{r}}$ over $\mathbb{F}_{q}$, and let coordinate-wise matrix products $*$ be defined via $\boldsymbol{\beta}$. For a primitive element $\gamma\in\mathbb{F}_{q^{r}}^{*}$, for $\mathbf{a}=(\gamma^{0},\gamma^{1},\ldots,\gamma^{g-1})\in\mathbb{F}_{q^{r}}^{g}$, and for any $k_{1},k_{2}=0,1,2,\ldots,N$, with $k_{1}\geq 1$, we have that

[TABLE]

if $k_{1}+k_{2}-1\leq N$, and $\mathcal{C}_{N,k_{1}}(\mathbf{a},\boldsymbol{\beta})*\mathcal{C}_{N,k_{2}}(\mathbf{a},\boldsymbol{\beta})=\mathbb{F}_{q^{r}}^{N}$ otherwise.

Proof.

It follows by combining Lemma 9, Theorem 2 and the fact that

[TABLE]

for all skew polynomials $F,G\in\mathbb{F}_{q^{r}}[x;\sigma]$. ∎

Setting $r=1$ and $\beta_{1}=1$, Corollary 11 recovers the well-known fact that the classical coordinate-wise product of two Reed-Solomon codes is again a Reed-Solomon code. See for instance [11, Prop. 3]. Setting $g=1$, Corollary 11 recovers the fact that the matrix product of two Gabidulin codes is again a Gabidulin codes. See for instance [23, Lemma 10] or [36].

5 Concrete PIR Schemes for LRS-based MR-LRC databases

In this section, we provide a concrete PIR scheme for the remaining MDS code of the MR-LRC storage code considered in Section 3, which is a linearized Reed-Solomon code (Definition 2). To that end, we will show how to construct the queries and how to reconstruct the file from the responses. The set of server responses, given the queries, are as described in Definition 5.

Let the notation be as in Section 3, fix an ordered basis $\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{r})\in\mathbb{F}_{q^{r}}^{r}$ of $\mathbb{F}_{q^{r}}$ over $\mathbb{F}_{q}$, and let coordinate-wise matrix products $*$ be defined via $\boldsymbol{\beta}$. We set throughout this section $t\geq 1$ as the target number of colluding servers, with the restriction

[TABLE]

and we will set $c=N-k-rt+1>0$.

For clarity, we present two schemes, being the first one (Subsection 5.1) a particular case of the second one (Subsection 5.2) by setting $b=1$. Both schemes will achieve the PIR rate

[TABLE]

coinciding with the rate of the PIR scheme in [11] for $(N,k)$ Reed-Solomon storage codes. See Section 1 for a discussion on the optimality of such rates. We also remark here that the scheme in [11] is recovered from our second scheme by setting $r=\delta=1$.

5.1 First Scheme: No Folding

Our first scheme assumes no minimum folding of the files, that is, $b=1$, and is precisely the particular case of our second scheme obtained by setting $b=1$. Note that the stored codewords may however be further folded $b^{\prime}\gg 1$ times without folding the PIR scheme. The main disadvantage of choosing $b=1$ is that the dimension $k$ must be divisible by $c=N-k-rt+1$. Relaxing this divisibility assumption is the advantage of the second scheme.

As stated above, assume that $k=sc$, for some $s\in\mathbb{N}$, which will be the number of iterations of the scheme. This can be trivially assumed if $k+rt=N$, thus $c=1$ and $s=k$. Note that this is the best choice for the competing parameters $k$ and $t$ satisfying (17), but it gives the smallest PIR rate $R=1/N\leq c/N$ among our schemes, although $R=1/N$ is still far better than downloading the whole database (which gives rate $1/m$), since $m\gg N$ in practice.

Fix file and iteration indices $i=1,2,\ldots,m$ and $u=1,$ $2,$ $\ldots,$ $s$, respectively. We now describe the two steps of $u$th iteration in Definition 5 to privately retrieve the $i$th file.

Step 1, Queries: Choose $m$ codewords $\mathbf{d}^{\ell}=(\mathbf{d}^{\ell}_{1},\mathbf{d}^{\ell}_{2},$ $\ldots,$ $\mathbf{d}^{\ell}_{g})\in\mathbb{F}_{q^{r}}^{N}$ uniformly at random from $\mathcal{C}_{N,rt}(\mathbf{a},\boldsymbol{\beta})$, where $\mathbf{d}^{\ell}_{j}\in\mathbb{F}_{q^{r}}^{r}$, for $\ell=1,2,\ldots,m$ and $j=1,2,\ldots,g$. The random vectors $\mathbf{d}^{\ell}=\mathbf{d}^{\ell}(u)$ depend on the iteration index $u$ (i.e. $\mathbf{d}^{\ell}(1),\mathbf{d}^{\ell}(2),\ldots,\mathbf{d}^{\ell}(s)$ are identically distributed but independent),s corres but we sometimes omit the index $u$ for simplicity. Set

[TABLE]

for $j=1,2,\ldots,g$. Define the set

[TABLE]

Finally, for each server $j=1,2,\ldots,g$, we define its query by

[TABLE]

where we define $\mathbf{e}_{j}^{i}(u)\in\mathbb{F}_{q^{r}}^{rm}$ as being zero everywhere except in the $i$th block of $r$ coordinates over $\mathbb{F}_{q^{r}}$, where it is defined as

[TABLE]

Here, we define $I_{J}\in\mathbb{F}_{q}^{r\times r}$, for a set $J\subseteq(j-1)r+[r]$, as the diagonal matrix $I_{J}=\operatorname{Diag}(\delta_{1}^{J},\delta_{2}^{J},\ldots,\delta_{r}^{J})$, where $\delta_{\kappa}^{J}=1$ if $(j-1)r+\kappa\in J$, and $\delta_{\kappa}^{J}=0$ otherwise. Note that $I_{\varnothing}\in\mathbb{F}_{q}^{r\times r}$ is the zero matrix.

Step 2, Responses: Due to the definition of the queries in (19) and the inner matrix product (13), the reader can check that the total response in the $u$th iteration is

[TABLE]

where $\mathbf{0}_{M}\in\mathbb{F}_{q^{r}}^{M}$ is a zero vector of length $M$.

File reconstruction: We now describe how to recover the $i$th file by combining the responses from all $s$ iterations. Let $H\in\mathbb{F}_{q^{r}}^{c\times N}$ be a parity-check matrix of

[TABLE]

(recall Corollary 11). For $u=1,$ $2,$ $\ldots,s$, we compute

[TABLE]

which holds since

[TABLE]

Since $\mathcal{C}_{N,k+rt-1}(\mathbf{a},\boldsymbol{\beta})$ is MDS by Theorem 1, its dual is also MDS, and we can recover the vector $\mathbf{z}^{i}_{J_{u}}\in\mathbb{F}_{q^{r}}^{c}$ from $\mathbf{r}^{i}(u)H^{T}$. Since we have that

[TABLE]

collecting all such $s$ restrictions $\mathbf{z}^{i}_{J_{u}}$, we obtain

[TABLE]

Now, since $\mathcal{C}_{N,k}(\mathbf{a},\boldsymbol{\beta})$ is MDS, again by Theorem 1, we may recover the $i$th file, $\mathbf{x}^{i}\in\mathbb{F}_{q^{r}}^{k}$, and we are done.

Note that the MDS property in this last step is not necessary: If we take the generator matrix $G_{out}$ of the outer code $\mathcal{C}_{N,k}(\mathbf{a},\boldsymbol{\beta})$ to be systematic, with the identity in the first $k$ columns, then it simply holds that $\mathbf{x}^{i}=(z^{i}_{1},z^{i}_{2},\ldots,z^{i}_{k})$.

Proof of privacy: We now show that the proposed PIR scheme protects against any $t$ colluding servers as in Definition 7. Recall from Section 3 that we identify servers with local groups. Let $T\subseteq[g]$, such that $|T|=t$, be the set of colluding local groups. Therefore, this can be understood as an adversary gaining as information the values $\mathbf{q}^{i}_{j}(u)\in\mathbb{F}_{q^{r}}^{rm}$, for $j\in T$, and for all iterations $u=1,2,\ldots,s$. We will just write $\mathbf{q}^{i}_{j}=\mathbf{q}^{i}_{j}(u)$ for simplicity. We need to prove that, for a given iteration, it holds that

[TABLE]

Since $\mathcal{C}_{N,rt}(\mathbf{a},\boldsymbol{\beta})\subseteq\mathbb{F}_{q^{r}}^{N}$ has dimension $rt$ and is MDS by Theorem 1, it holds that any set of $rt$ coordinates in $[N]$ constitute an information set for $\mathcal{C}_{N,rt}(\mathbf{a},\boldsymbol{\beta})$. In other words, the restricted code $\mathcal{C}_{N,rt}(\mathbf{a},\boldsymbol{\beta})_{\widetilde{T}}=\mathbb{F}_{q^{r}}^{rt}$ is the whole space, where $\widetilde{T}=\bigcup_{j\in T}((j-1)r+[r])\subseteq[N]$ is the actual set of colluding nodes. This implies that the vectors

[TABLE]

are uniform random variables in $\mathbb{F}_{q^{r}}^{rt}$. Since the Cartesian product of uniform random variables is again a uniform random variable, we deduce that

[TABLE]

is a uniform random variable in $\mathbb{F}_{q^{r}}^{rtm}$. Since the vector of queries $(\mathbf{q}_{j}^{i})_{j\in T}$ is a translation of the random variable $(\mathbf{d}_{j})_{j\in T}$ by a deterministic vector, we deduce that $(\mathbf{q}_{j}^{i})_{j\in T}$ is a uniform random variable in $\mathbb{F}_{q^{r}}^{rtm}$. Since there is only one uniform random variable in $\mathbb{F}_{q^{r}}^{rtm}$, independently of $i$, we deduce that $I((\mathbf{q}^{i}_{j})_{j\in T};i)=0$, and we are done.

5.2 Second Scheme: Folding

In our second scheme, we avoid the constraint that $k$ must be divisible by $c=N-k-rt+1$. To that end, we will make use of the folding parameter $b$ as done in [11]. Again, the stored codewords may be further folded $b^{\prime}\gg 1$ times without further folding the PIR scheme. We emphasize here that the scheme in [11] is actually recovered from this second scheme by setting $r=\delta=1$, which is the case in which linearized Reed-Solomon codes recover Reed-Solomon codes (see Subsection 2.1). Our first scheme is also recovered from this second scheme by setting $b=1$. To avoid the divisibility assumption, we define

[TABLE]

hence guaranteeing that $bk=sc$. Thus we may define

[TABLE]

Fix file and iteration indices $i=1,2,\ldots,m$ and $u=1,$ $2,$ $\ldots,$ $s$, respectively. We now describe the two steps of $u$th iteration in Definition 5 to privately retrieve the $i$th file.

Step 1, Queries: Choose $mb$ codewords $\mathbf{d}^{\ell,v}=(\mathbf{d}^{\ell,v}_{1},$ $\mathbf{d}^{\ell,v}_{2},$ $\ldots,$ $\mathbf{d}^{\ell,v}_{g})\in\mathbb{F}_{q^{r}}^{N}$, uniformly at random from $\mathcal{C}_{N,rt}(\mathbf{a},\boldsymbol{\beta})$, where $\mathbf{d}^{\ell,v}_{j}\in\mathbb{F}_{q^{r}}^{r}$, for $\ell=1,2,\ldots,m$, $v=1,$ $2,$ $\ldots,b$, and $j=1,2,\ldots,g$. As before, $\mathbf{d}^{\ell,v}=\mathbf{d}^{\ell,v}(u)$ depends on $u$, but we sometimes drop this in the notation. We set

[TABLE]

[TABLE]

for $\ell=1,2,\ldots,m$ and $j=1,2,\ldots,g$. Define the sets

[TABLE]

Finally, for each server $j=1,2,\ldots,g$, we define its query by

[TABLE]

In this case, we define $\mathbf{e}_{j}^{i}(u)\in\mathbb{F}_{q^{r}}^{rbm}$ as being zero everywhere except in the $(b(i-1)+v)$th block of $r$ coordinates over $\mathbb{F}_{q^{r}}$, for $v=1,2,\ldots,b$, where it is defined as

[TABLE]

As before, we define $I_{J}\in\mathbb{F}_{q}^{r\times r}$, for a set $J\subseteq(j-1)r+[r]$, as the diagonal matrix $I_{J}=\operatorname{Diag}(\delta_{1}^{J},\delta_{2}^{J},\ldots,\delta_{r}^{J})$, where $\delta_{\kappa}^{J}=1$ if $(j-1)r+\kappa\in J$, and $\delta_{\kappa}^{J}=0$ otherwise. As before, $I_{\varnothing}\in\mathbb{F}_{q}^{r\times r}$ is the zero matrix.

Step 2, Responses: The reader can check that, from the definition of the queries in (21) and the inner matrix product (13), the total response in the first iteration is

[TABLE]

where $\mathbf{0}$ has length $N-c$. In the $u$th iteration, the response is obtained similarly, replacing $\mathbf{z}_{J_{1}^{v}}^{i,v}$ by $\mathbf{z}_{J_{u}^{v}}^{i,v}$, but placed in the coordinates indexed by $J_{u}^{v}$ taking the cyclicity of the coordinates in $[N]$ into account, for $v=1,2,\ldots,b$.

File reconstruction: We now describe how to recover the $i$th file by combining the responses from all $s$ iterations. As before, let $H\in\mathbb{F}_{q^{r}}^{c\times N}$ be a parity-check matrix of

[TABLE]

(recall Corollary 11). In the first iteration, we compute

[TABLE]

which holds since

[TABLE]

As before, $\mathcal{C}_{N,k+rt-1}(\mathbf{a},\boldsymbol{\beta})$ is MDS by Theorem 1, and thus its dual is also MDS. Therefore we may recover $\mathbf{z}_{J_{1}^{1}}^{i,1},$ $\mathbf{z}_{J_{1}^{2}}^{i,2},$ $\ldots,\mathbf{z}_{J_{1}^{b}}^{i,b}\in\mathbb{F}_{q^{r}}^{h}$ from $\mathbf{r}^{i}H^{T}$.

In a similar way, in the $u$th iteration, we recover $\mathbf{z}_{J_{u}^{1}}^{i,1},$ $\mathbf{z}_{J_{u}^{2}}^{i,2},$ $\ldots,\mathbf{z}_{J_{u}^{b}}^{i,b}\in\mathbb{F}_{q^{r}}^{h}$, for $u=1,2,\ldots,s$. For a given $v=1,$ $2,$ $\ldots,$ $b$, the reader can check from their definition that the sets $J_{1}^{v},J_{2}^{v},\ldots,J_{s}^{v}$ are disjoint and the size of their union is $sh=k$. Therefore, we recover $k$ symbols of $\mathbf{z}^{i,v}\in\mathbb{F}_{q^{r}}^{N}$ over the alphabet $\mathbb{F}_{q^{r}}$, together with their indices, given by $J_{1}^{v}\cup J_{2}^{v}\cup\ldots\cup J_{s}^{v}\subseteq[N]$. Since $\mathcal{C}_{N,k}(\mathbf{a},\boldsymbol{\beta})$ is MDS, we recover the $v$th row of the $i$th file, that is, $\mathbf{x}^{i,v}\in\mathbb{F}_{q^{r}}^{k}$, for $v=1,2,\ldots,b$. Thus we are done by collecting all $b$ rows, $\mathbf{x}^{i,1},\mathbf{x}^{i,2},\ldots,\mathbf{x}^{i,b}$, of the $i$th file.

Proof of privacy: Analogous to that in Subsection 5.1.

6 Further Considerations

6.1 Unequal Localities and Local Distances

The results in this work may be extended, in a straightforward way, to the case where each local group $\Gamma_{j}$ has a different locality $r_{j}$ and local distance $\delta_{j}$, for $j=1,2,\ldots,g$. See the next subsection for a further extension. The MR-LRC in Construction 1 based on linearized Reed-Solomon codes can be extended to arbitrary equal or unequal localities and local distances as long as the field is $\mathbb{F}_{q^{r}}$, where $q>g$ and $r\geq\max\{r_{1},r_{2},\ldots,r_{g}\}$. See [22, Sec. III]. By choosing systematic generator matrices of the MDS local linear codes (which now are different), the remaining MDS storage code after removing all local redundancies is again a $k$-dimensional linearized Reed-Solomon code, although of length $N=\sum_{j=1}^{g}r_{j}$.

For $\tau\geq 1$ colluding nodes, the achieved rate would still be $R=(N-k-\tau+1)/N$. However, $t\geq 1$ colluding local groups correspond in this case to a number of colluding nodes that is different for different sets of local groups. In other words, the collusion pattern [33] is generated by maximal collusion sets of different sizes. We may still proceed with the strategy in this work, that is, we may consider protecting against any $\tau=\max\{\sum_{j\in T}r_{j}\mid T\subseteq[g],|T|=t\}$ colluding nodes. Improvements on the rate $R=(N-k-\tau+1)/N$ may be possible for certain cases (as in [33, Sec. V]), which we leave open.

Finally, the main motivation behind unequal localities and local distances is that some local groups may require faster and/or more robust repair, for instance due to hot data, while global erasure correction may be improved by considering the different localities and local distances. See [5, 16, 38] for more details.

6.2 Arbitrary Local Linear Codes and Hierarchical Localities

As before, the results in this work may be extended, in a straightforward way, to the case where each local group $\Gamma_{j}$ uses an arbitrary $r_{j}$-dimensional local linear code $\mathcal{C}_{loc}^{j}\subseteq\mathbb{F}_{q}^{n_{j}}$, where $\Gamma_{j}=|n_{j}|$ and $r_{j}+\delta_{j}-1\leq n_{j}$, where $\delta_{j}=\operatorname{d}(\mathcal{C}_{loc}^{j})$, for $j=1,2,\ldots,g$. Construction 1 still gives an MR-LRC for any choice of local linear codes, see [22, Sec. IV]. Furthermore, the local codes may be dynamically, efficiently and locally updated in order to adapt to different distributed storage configurations, as discussed in [22, Subsec. V-A]. In particular, the local codes may be in turn MR-LRCs, giving rise to multi-layer or hierarchical MR-LRCs (see [22, Def. 7], [22, Subsec. V-B] and [24, Sec. II]), which have optimal global distance by [22, Th. 4]. As before, by choosing systematic generator matrices of the local codes, the remaining MDS storage code after removing the local redundancies is a linearized Reed-Solomon code of length $N=\sum_{j=1}^{g}r_{j}$.

6.3 PIR over Linearly Coded Networks

Linear network coding [19] permits maximum information flow over a network from a source to several sinks simultaneously in one shot (multicast). In [36], PIR is considered where each server is formed by a number $r\geq 1$ of nodes in the database and communication between the user and each server is through a linearly coded network.

To avoid mixing information through the network for non-colluding sets of servers, it is assumed in [36] that the linearly coded networks between the user and the servers are pair-wise disjoint (after removing the user node). This makes the total transfer matrix from the user to the database and back have a block-diagonal shape $\operatorname{Diag}(A_{1},A_{2},\ldots,A_{g})\in\mathbb{F}_{q}^{gr}$, where $A_{j}\in\mathbb{F}_{q^{r}}^{r}$ is the transfer matrix from the $j$th server to the user (we assume square transfer matrices for simplicity). In other words, the total linearly coded network from the user to the servers and back can be considered as a multishot linearly coded network as in [21], with one shot per server. The effect of such a channel is simply multiplying codewords by $\operatorname{Diag}(A_{1},A_{2},\ldots,A_{g})$.

It was shown in [21, Subsec. V-F] that multiplying on the right a linearized Reed-Solomon code, as in Definition 2, by a block-diagonal matrix $\operatorname{Diag}(A_{1},A_{2},\ldots,A_{g})\in\mathbb{F}_{q}^{gr}$ gives again a linearized Reed-Solomon code, possibly with erasures if the matrices $A_{j}$ are not full-rank. Using this fact, our PIR scheme (Subsection 5.2) may be used mutatis mutandis in the scenario described in this subsection and in [36]. In the error-free and erasure-free case, the rate obtained in both works is

[TABLE]

However, since Gabidulin codes [12] are used in [36], the required field size is $q_{0}^{gr}$, where $q_{0}\geq 2$ is the field size of the underlying linear network code. Note that $q_{0}^{gr}$ is exponential in the number of servers $g$, whereas our scheme would still require the field size $g^{r}$, which is polynomial in the number of servers $g$.

Acknowledgement

The author gratefully acknowledges the support from The Independent Research Fund Denmark (Grant No. DFF-7027-00053B).