This paper revisits the problem of multi-server Private Information Retrieval with Private Side Information (PIR-PSI). In this problem, $N$ non-colluding servers store identical copies of $K$ messages, each comprising $L$ symbols from $\mathbb{F}_q$, and a user, who knows $M$ of these messages, wants to retrieve one of the remaining $K-M$ messages. The user's goal is to retrieve the desired message by downloading the minimum amount of information from the servers while revealing no information about the identities of the desired message and side information messages to any server. The capacity of PIR-PSI, defined as the maximum achievable download rate, was previously characterized for all $N$, $K$, and $M$ when $L$ and $q$ are sufficiently large -- specifically, growing exponentially with $K$, to ensure the divisibility of each message into $N^K$ sub-packets and to guarantee the existence of an MDS code with its length and dimension being exponential in $K$. In this work, we propose a new capacity-achieving PIR-PSI scheme that is applicable to all $N$, $K$, $M$, $L$, and $q$ where $N\geq M+1$ and $N-1\mid L$. The proposed scheme operates with a sub-packetization level of $N-1$, independent of $K$, and works over any finite field without requiring an MDS code.
Suppose a database containing $M$ records is replicated across $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a private information retrieval (PIR) scheme. In practice, capacity-achieving and small sub-packetization are both desired for PIR schemes, because the former implies the highest download rate and the latter usually means simple realization. For general values of $N,T,M$, the only known capacity-achieving PIR scheme wa…
Consider the problem of private information retrieval (PIR) over a distributed storage system where $M$ records are stored across $N$ servers by using an $[N,K]$ MDS code. For simplicity, this problem is usually referred as the coded PIR problem. In 2016, Banawan and Ulukus designed the first capacity-achieving coded PIR scheme with sub-packetization $KN^{M}$ and access number $MKN^{M}$, where capacity characterizes the minimal download size for retrieving per unit of data, and sub-packetization and access number are two metrics closely related to implementation complexity. In this paper, we f…
In the literature, all the known high-rate MDS codes with the optimal repair bandwidth possess a significantly large sub-packetization level, which may prevent the codes to be implemented in practical systems. To build MDS codes with small sub-packetization level, existing constructions and theoretical bounds imply that one may sacrifice the optimality of the repair bandwidth. Partly motivated by the work of Tamo et al. (IEEE Trans. Inform. Theory, 59(3), 1597-1616, 2013), in this paper, we present a transformation that can greatly reduce the sub-packetization level of MDS codes with the optim…
Suppose a database containing $M$ records is replicated across $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$-private information retrieval ($T$-PIR) scheme. Three indexes are concerned for PIR schemes: (1)rate, indicating the amount of retrieved information per unit of downloaded data. The highest achievable rate is characterized by the capacity; (2) sub-packetization, reflexing the implementation complexity for linear …
An $(n,k)$ maximum distance separable (MDS) code has optimal repair access if the minimum number of symbols accessed from $d$ surviving nodes is achieved, where $k+1\le d\le n-1$. Existing results show that the sub-packetization $α$ of an $(n,k,d)$ high code rate (i.e., $k/n>0.5$) MDS code with optimal repair access is at least $(d-k+1)^{\lceil\frac{n}{d-k+1}\rceil}$. In this paper, we propose a class of multi-layer transformed MDS codes such that the sub-packetization is $(d-k+1)^{\lceil\frac{n}{(d-k+1)η}\rceil}$, where $η=\lfloor\frac{n-k-1}{d-k}\rfloor$, and the repair access is optimal for…
This paper investigates reducing sub-packetization of capacity-achieving schemes for uncoded Storage Constrained Private Information Retrieval (SC-PIR) systems. In the SC-PIR system, a user aims to retrieve one out of $K$ files from $N$ servers while revealing nothing about its identity to any individual server, in which the $K$ files are stored at the $N$ servers in an uncoded form and each server can store up to $μK$ equivalent files, where $μ$ is the normalized storage capacity of each server. We first prove that there exists a capacity-achieving SC-PIR scheme for a given storage design if …
An explicit construction of systematic MDS codes, called HashTag+ codes, with arbitrary sub-packetization level for all-node repair is proposed. It is shown that even for small sub-packetization levels, HashTag+ codes achieve the optimal MSR point for repair of any parity node, while the repair bandwidth for a single systematic node depends on the sub-packetization level. Compared to other codes in the literature, HashTag+ codes provide from 20% to 40% savings in the average amount of data accessed and transferred during repair.
This paper addresses the problem of constructing MDS codes that enable exact repair of each code block with small repair bandwidth, which refers to the total amount of information flow from the remaining code blocks during the repair process. This problem naturally arises in the context of distributed storage systems as the node repair problem [7]. The constructions of exact-repairable MDS codes with optimal repair-bandwidth require working with large sub-packetization levels, which restricts their employment in practice. This paper presents constructions for MDS codes that simultaneously prov…
An $(n, M)$ vector code $\mathcal{C} \subseteq \mathbb{F}^n$ is a collection of $M$ codewords where $n$ elements (from the field $\mathbb{F}$) in each of the codewords are referred to as code blocks. Assuming that $\mathbb{F} \cong \mathbb{B}^{\ell}$, the code blocks are treated as $\ell$-length vectors over the base field $\mathbb{B}$. Equivalently, the code is said to have the sub-packetization level $\ell$. This paper addresses the problem of constructing MDS vector codes which enable exact reconstruction of each code block by downloading small amount of information from the remaining code …
An $(n,k,l)$ MDS array code of length $n,$ dimension $k=n-r$ and sub-packetization $l$ is formed of $l\times n$ matrices over a finite field $F,$ with every column of the matrix stored on a separate node in a distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node can be performed by accessing a set of $d\le n-1$ helper nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with $d=n-1,$ the sub-packetization $l$ satisfies the…
Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education（教育部密码技术与信息安全重点实验室）；School of Cyber Science and Technology, Shandong University（山东大学网络空间安全学院）；Quan Cheng Laboratory（泉城实验室）
We propose a private information retrieval (PIR) protocol for distributed storage systems with noncolluding nodes where data is stored using an arbitrary linear code. An expression for the PIR rate, i.e., the ratio of the amount of retrieved data per unit of downloaded data, is derived, and a necessary and a sufficient condition for codes to achieve the maximum distance separable (MDS) PIR capacity are given. The necessary condition is based on the generalized Hamming weights of the storage code, while the sufficient condition is based on code automorphisms. We show that cyclic codes and Reed-…
We consider private information retrieval (PIR) for distributed storage systems (DSSs) with noncolluding nodes where data is stored using a non maximum distance separable (MDS) linear code. It was recently shown that if data is stored using a particular class of non-MDS linear codes, the MDS-PIR capacity, i.e., the maximum possible PIR rate for MDS-coded DSSs, can be achieved. For this class of codes, we prove that the PIR capacity is indeed equal to the MDS-PIR capacity, giving the first family of non-MDS codes for which the PIR capacity is known. For other codes, we provide asymmetric PIR pr…
We consider private information retrieval (PIR) for distributed storage systems (DSSs) with noncolluding nodes where data is stored using a non maximum distance separable (MDS) linear code. It was recently shown that if data is stored using a particular class of non-MDS linear codes, the MDS-PIR capacity, i.e., the maximum possible PIR rate for MDS-coded DSSs, can be achieved. For this class of codes, we prove that the PIR capacity is indeed equal to the MDS-PIR capacity, giving the first family of non-MDS codes for which the PIR capacity is known. For other codes, we provide asymmetric PIR pr…
The capacity of private information retrieval (PIR) from databases coded using maximum distance separable (MDS) codes has been previously characterized by Banawan and Ulukus, where it was assumed that the messages are encoded and stored separably into the databases. This assumption was also usually taken in other related works in the literature, and this capacity is usually referred to as the MDS-PIR capacity colloquially. In this work, we considered the question if and when this capacity barrier can be broken through joint encoding and storing of the messages. Our main results are two classes…
Private information retrieval (PIR) is the problem of privately retrieving one out of $M$ original files from $N$ severs, i.e., each individual server learns nothing about the file that the user is requesting. Usually, the $M$ files are replicated or encoded by a maximum distance separable (MDS) code and then stored across the $N$ servers. Compared to mere replication, MDS coded servers can significantly reduce the storage overhead. Particularly, PIR from minimum storage regenerating (MSR) coded servers can simultaneously reduce the repair bandwidth when repairing failed servers. Existing PIR …
This paper presents an explicit construction for an $((n,k,d=n-1), (α,β))$ regenerating code over a field $\mathbb{F}_Q$ operating at the Minimum Storage Regeneration (MSR) point. The MSR code can be constructed to have rate $k/n$ as close to $1$ as desired, sub-packetization given by $r^{\frac{n}{r}}$, for $r=(n-k)$, field size no larger than $n$ and where all code symbols can be repaired with the same minimum data download. The construction modifies a prior construction by Sasidharan et. al. which required far larger field-size. A building block appearing in the construction is a scalar MDS …
Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. Specifically, an $(n, k)$ MDS code stores $k$ symbols in $n$ disks such that the overall system is tolerant to a failure of up to $n-k$ disks. However, access to at least $k$ disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length $\ell$. MDS array codes have the potential to repair a single erasure using a fraction $1/(n-k)$ of data stored in the remaining disks. We introduce new methods of a…
For high-rate maximum distance separable (MDS) codes, most of them are designed to optimally repair a single failed node by connecting all the surviving nodes. However, in practical systems, sometimes not all the surviving nodes are available. To facilitate the practical storage system, a few constructions of $(n,k)$ MDS codes with the property that any single failed node can be optimally repaired by accessing any $d$ surviving nodes have been proposed, where $d\in [k+1:n-1)$. However, all of them either have large sub-packetization levels or are not explicit for all the parameters. To address…
Private information retrieval (PIR) gets renewed attentions due to its information-theoretic reformulation and its application in distributed storage system (DSS). The general PIR model considers a coded database containing $N$ servers storing $M$ files. Each file is stored independently via the same arbitrary $(N,K)$-MDS code. A user wants to retrieve a specific file from the database privately against an arbitrary set of $T$ colluding servers. A key problem is to analyze the PIR capacity, defined as the maximal number of bits privately retrieved per one downloaded bit. Several extensions for…
A Disguise-and-Squeeze PIR Scheme for the MDS-TPIR Setting and Beyond
State Key Laboratory of Cryptography and Digital Economy Security, Key Laboratory of Cryptologic Technology and Information Security of Ministry of Education, School of Cyber Science and Technology, Shandong University（山东大学网络空间科学学院，密码与数字经济安全全国重点实验室，教育部密码技术信息安全重点实验室）；School of Information Science and Engineering, Shandong Agricultural University（山东农业大学信息科学与工程学院）
We prove that a class of distance-optimal local reconstruction codes (LRCs), an important family of repair-efficient codes for distributed storage systems, achieve the maximum distance separable private information retrieval capacity for the case of noncolluding nodes. This particular class of codes includes Pyramid codes and other LRCs proposed in the literature.
Suppose a database containing $M$ records is replicated in each of $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$-private information retrieval ($T$-PIR) scheme. In this paper we focus on the field size of $T$-PIR schemes. We design a generalcapacity-achieving $T$-PIR scheme whose queries are generated by using some {\rm MDS } array codes. It only requires field size $q\geq\sqrt[\ell]{N}$, where $\ell=\min\{t^{M-2},(n-t)…
We propose a generic transformation that can convert any nonbinary $(n=k+r,k)$ maximum distance separable (MDS) code into another $(n,k)$ MDS code over the same field such that 1) some arbitrarily chosen $r$ nodes have the optimal repair bandwidth and the optimal rebuilding access, 2) for the remaining $k$ nodes, the normalized repair bandwidth and the normalized rebuilding access (over the file size) are preserved, 3) the sub-packetization level is increased only by a factor of $r$. Two immediate applications of this generic transformation are then presented. The first application is that we …
Foshan University（佛山大学）；Chern Institute of Mathematics and LPMC, Nankai University（南开大学陈省身数学研究所和理论物理研究中心）；School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen（哈尔滨工业大学深圳计算机科学与技术学院）
We present a high-rate $(n,k,d=n-1)$-MSR code with a sub-packetization level that is polynomial in the dimension $k$ of the code. While polynomial sub-packetization level was achieved earlier for vector MDS codes that repair systematic nodes optimally, no such MSR code construction is known. In the low-rate regime (i. e., rates less than one-half), MSR code constructions with a linear sub-packetization level are available. But in the high-rate regime (i. e., rates greater than one-half), the known MSR code constructions required a sub-packetization level that is exponential in $k$. In the pres…
Reducing The Sub-packetization Level of Optimal-Access Cooperative MSR Codes
Ministry of Education Key Laboratory of Mathematics and Information Networks, School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学与信息技术教育部重点实验室）；School of Information Science and Engineering, Shandong Agricultural University（山东农业大学信息科学与工程学院）
In the literature, most of the known high-rate $(n,k)$ MDS array codes with the optimal repair property only support a single repair degree (i.e., the number of helper nodes contacted during a repair process) $d$, where $k\le d\le n-1$. However, in practical storage systems, the number of available nodes changes frequently. Thus, it is preferred to construct $(n,k)$ MDS array codes with multiple repair degrees and the optimal repair property for all nodes. To the best of our knowledge, only two MDS array codes have such properties in the literature, which were proposed by Ye and Barg (IEEE Tra…
Accessing the data in the failed disk (degraded read) with low latency is crucial for an erasure-coded storage system. In this work, the maximum distance separable (MDS) array code with the property of degraded-read friendly (DRF) is discussed. For the DRF MDS array code with 2 redundant nodes and the sub-packetization level of 2, the lower bound of its access bandwidth is derived.
Private Information Retrieval from Joint Systematic MDS-Coded with Non-Colluding Servers: Bounds and Constructions
School of Information Science and Engineering, Shandong Agricultural University（山东农业大学信息科学与工程学院）；State Key Laboratory of Cryptography and Digital Economy Security, Shandong University（山东大学密码与数字经济安全国家重点实验室）；Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University（山东大学教育部密码技术与信息安全重点实验室）；School of Cyber Science and Technology, Shandong University（山东大学网络空间安全学院）
In this work, a flexible and robust private information retrieval (PIR) scheme based on binary non-maximum distance separable (non-MDS) codes is considered. This combines previous works on PIR schemes based on transitive non-MDS codes on one hand, and PIR from MDS-coded Byzantine and non-responsive servers on the other hand. More specifically, a PIR scheme employing binary Reed-Muller (RM) codes tolerant to colluding, Byzantine, and non-responsive servers is constructed, and bounds for the achievable rates are derived under certain conditions. The construction of such schemes turns out to be m…
We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs) where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR ra…
The problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS), is considered. The DSS is coded by an $(n,k,d)$ Maximum Distance Separable (MDS) code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing, to the spy nodes, no information on which data is being requested. A user …
Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{Ω(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $ε$-MSR codes, which incur a factor of $(1+ε)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes…
Piggybacking codes are a special class of MDS array codes that can achieve small repair bandwidth with small sub-packetization by first creating some instances of an $(n,k)$ MDS code, such as a Reed-Solomon (RS) code, and then designing the piggyback function. In this paper, we propose a new piggybacking coding design which designs the piggyback function over some instances of both $(n,k)$ MDS code and $(n,k')$ MDS code, when $k\geq k'$. We show that our new piggybacking design can significantly reduce the repair bandwidth for single-node failures. When $k=k'$, we design piggybacking code that…
As a special class of array codes, $(n,k,m)$ piggybacking codes are MDS codes (i.e., any $k$ out of $n$ nodes can retrieve all data symbols) that can achieve low repair bandwidth for single-node failure with low sub-packetization $m$. In this paper, we propose two new piggybacking codes that have lower repair bandwidth than the existing piggybacking codes given the same parameters. Our first piggybacking codes can support flexible sub-packetization $m$ with $2\leq m\leq n-k$, where $n - k > 3$. We show that our first piggybacking codes have lower repair bandwidth for any single-node failure th…
Partial maximum distance separable (PMDS) codes are a kind of erasure codes where the nodes are divided into multiple groups with each forming an MDS code with a smaller code length, thus they allow repairing a failed node with only a few helper nodes and can correct all erasure patterns that are information-theoretically correctable. However, the repair of a failed node of PMDS codes still requires a large amount of communication if the group size is large. Recently, PMDS array codes with each local code being an MSR code were introduced to reduce the repair bandwidth further. However, they r…
This paper presents private information retrieval (PIR) schemes for coded storage with colluding servers, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. This rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems without collusion. While many PIR schemes in the literature require field sizes that grow with the num…
The problem of private information retrieval gets renewed attentions in recent years due to its information-theoretic reformulation and applications in distributed storage systems. PIR capacity is the maximal number of bits privately retrieved per one bit of downloaded bit. The capacity has been fully solved for some degenerating cases. For a general case where the database is both coded and colluded, the exact capacity remains unknown. We build a general private information retrieval scheme for MDS coded databases with colluding servers. Our scheme achieves the rate $(1+R+R^2+\cdots+R^{M-1})$…
We consider the problem of Private Information Retrieval with Private Side Information (PIR-PSI), wherein a user wants to retrieve a file from replication based non-colluding databases by using the prior knowledge of a subset of the files stored on the databases. The PIR-PSI framework ensures that the privacy of the demand and the side information are jointly preserved, thereby finding potential applications when multiple files have to be downloaded spread across different time-instants. Although the capacity of the PIR-PSI setting is known, we observe that the underlying capacity achieving co…
We study private information retrieval (PIR) on coded data with possibly colluding servers. Devising PIR schemes with optimal download rate in the case of collusion and coded data is still open in general. We provide a lifting operation that can transform what we call one-shot PIR schemes for two messages into schemes for any number of messages. We apply this lifting operation on existing PIR schemes and describe two immediate implications. First, we obtain novel PIR schemes with improved download rate in the case of MDS coded data and server collusion. Second, we provide a simplified descript…
We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from $N$ non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any $T$ databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when $K>T/\textbf{gcd}(N,T)$ where $K$ is the total number of messages in the system and $\textbf…
We study the performance of Reed-Solomon (RS) codes for the \em exact repair problem \em in distributed storage. Our main result is that, in some parameter regimes, Reed-Solomon codes are optimal regenerating codes, among MDS codes with linear repair schemes. Moreover, we give a characterization of MDS codes with linear repair schemes which holds in any parameter regime, and which can be used to give non-trivial repair schemes for RS codes in other settings. More precisely, we show that for $k$-dimensional RS codes whose evaluation points are a finite field of size $n$, there are exact repair …
An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading $\ell/r$ field elements (which is known to be the least possible) from each of the other symbols. MSR codes are attractive for use in…
We consider the problem of private information retrieval (PIR) from multiple storage nodes when the underlying database is encoded using regenerating codes, i.e., the database has the ability to recover from individual node failures. We seek optimality in terms of download rates when a user wants to retrieve a message privately (in an information-theoretic sense) from the database. We focus mostly on the more extensively explored Minimum Storage Regenerating (MSR) codes and formally state and prove the PIR capacity of such codes. We also give a simple scheme to achieve this capacity which is b…
In quantum private information retrieval (QPIR), a user retrieves a classical file from multiple servers by downloading quantum systems without revealing the identity of the file. The QPIR capacity is the maximal achievable ratio of the retrieved file size to the total download size. In this paper, the capacity of QPIR from MDS-coded and colluding servers is studied for the first time. Two general classes of QPIR, called stabilizer QPIR and dimension-squared QPIR induced from classical strongly linear PIR are defined, and the related QPIR capacities are derived. For the non-colluding case, the…
Private Information Retrieval (PIR) was first proposed by B. Chor, O. Goldreich, E. Kushilevitz and M. Sudan in their 1995 FOCS paper. For MDS coded distributed storage system private information retrieval was proposed and the capacity of PIR schemes for MDS coded distributed storage was studied. Star product PIR schemes from general coded distributed storage system with colluding servers were constructed over general finite fields. These star product schemes has no restriction on the sizes of fields and can be constructed for coded distributed storage across large number of servers. In this p…
We consider the problem of multiple-node repair in distributed storage systems under the cooperative model, where the repair bandwidth includes the amount of data exchanged between any two different storage nodes. Recently, explicit constructions of MDS codes with optimal cooperative repair bandwidth for all possible parameters were given by Ye and Barg (IEEE Transactions on Information Theory, 2019). The node size (or sub-packetization) in this construction scales as $\exp(Θ(n^h))$, where $h$ is the number of failed nodes and $n$ is the code length. In this paper, we give new explicit constru…
This paper presents a construction for high-rate MDS codes that enable bandwidth-efficient repair of a single node. Such MDS codes are also referred to as the minimum storage regenerating (MSR) codes in the distributed storage literature. The construction presented in this paper generates MSR codes for all possible number of helper nodes $d$ as $d$ is a design parameter in the construction. Furthermore, the obtained MSR codes have polynomial sub-packetization (a.k.a. node size) $α$. The construction is built on the recent code proposed by Sasidharan et al. [1], which works only for $d = n-1$, …
