This paper considers the problem of multi-server Private Linear Computation, under the joint and individual privacy guarantees. In this problem, identical copies of a dataset comprised of $K$ messages are stored on $N$ non-colluding servers, and a user wishes to obtain one linear combination of a $D$-subset of messages belonging to the dataset. The goal is to design a scheme for performing the computation such that the total amount of information downloaded from the servers is minimized, while the privacy of the $D$ messages required for the computation is protected. When joint privacy is required, the identities of all of these $D$ messages must be kept private jointly, and when individual privacy is required, the identity of every one of these $D$ messages must be kept private individually. In this work, we characterize the capacity, which is defined as the maximum achievable download rate, under both joint and individual privacy requirements. In particular, we show that when joint privacy is required the capacity is given by ${(1+1/N+\dots+1/N^{K-D})^{-1}}$, and when individual privacy is required the capacity is given by ${(1+1/N+\dots+1/N^{\lceil K/D\rceil-1})^{-1}}$ assuming that $D$ divides $K$, or $K\pmod D$ divides $D$. Our converse proofs are based on reduction from two variants of the multi-server Private Information Retrieval problem in the presence of side information. Our achievability schemes build up on our recently proposed schemes for single-server Private Linear Transformation and the multi-server private computation scheme proposed by Sun and Jafar. Using similar proof techniques, we also establish upper and lower bounds on the capacity for the cases in which the user wants to compute $L$ (potentially more than one) linear combinations.
Private Computation (PC), recently introduced by Sun and Jafar, is a generalization of Private Information Retrieval (PIR) in which a user wishes to privately compute an arbitrary function of data stored across several servers. We construct a PC scheme which accounts for server collusion, coded data, and non-linear functions. For data replicated over several possibly colluding servers, our scheme computes arbitrary functions of the data with rate equal to the asymptotic capacity of PIR for this setup. For systematically encoded data stored over colluding servers, we privately compute arbitrary…
In order to preserve the privacy of the users demands from other users, in this paper we formulate a novel information theoretic Device-to-Device (D2D) private caching model by adding a trusted server. In the delivery phase, the trusted server collects the users demands and sends a query to each user, who then broadcasts packets according to this query. Two D2D private caching schemes (uncoded and coded) are proposed in this paper, which are shown to be order optimal.
This paper considers the problem of single-server Individually-Private Information Retrieval with side information (IPIR). In this problem, there is a remote server that stores a dataset of $K$ messages, and there is a user that initially knows $M$ of these messages, and wants to retrieve $D$ other messages belonging to the dataset. The goal of the user is to retrieve the $D$ desired messages by downloading the minimum amount of information from the server while revealing no information about whether an individual message is one of the $D$ desired messages. In this work, we focus on linear IPI…
We study the problem of single-server single-message Private Information Retrieval with Private Coded Side Information (PIR-PCSI). In this problem, there is a server that stores a database, and a user who knows a random linear combination of a random subset of messages in the database. The number of messages contributing to the user's side information is known to the server a priori, whereas their indices and coefficients are unknown to the server a priori. The user wants to retrieve a message from the server (with minimum download cost), while protecting the identities of both the demand and …
In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. Further, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a generic point of view.
This paper focuses on the Private Linear Transformation (PLT) problem in the multi-server scenario. In this problem, there are $N$ servers, each of which stores an identical copy of a database consisting of $K$ independent messages, and there is a user who wishes to compute $L$ independent linear combinations of a subset of $D$ messages in the database while leaking no information to the servers about the identity of the entire set of these $D$ messages required for the computation. We focus on the setting in which the coefficient matrix of the desired $L$ linear combinations generates a Maxim…
We present a general framework for Private Information Retrieval (PIR) from arbitrary coded databases, that allows one to adjust the rate of the scheme according to the suspected number of colluding servers. If the storage code is a generalized Reed-Solomon code of length n and dimension k, we design PIR schemes which simultaneously protect against t colluding servers and provide PIR rate 1-(k+t-1)/n, for all t between 1 and n-k. This interpolates between the previously studied cases of t=1 and k=1 and asymptotically achieves the known capacity bounds in both of these cases, as the size of the…
In this paper, we consider the multi-server setting of Private Information Retrieval with Private Coded Side Information (PIR-PCSI) problem. In this problem, there is a database of $K$ messages whose copies are replicated across $N$ servers, and there is a user who knows a random linear combination of a random subset of $M$ messages in the database as side information. The user wishes to download one message from the servers, while protecting the identities of both the demand message and the messages forming the side information. We assume that the servers know the number of messages forming t…
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…
We introduce the problem of private computation, comprised of $N$ distributed and non-colluding servers, $K$ independent datasets, and a user who wants to compute a function of the datasets privately, i.e., without revealing which function he wants to compute, to any individual server. This private computation problem is a strict generalization of the private information retrieval (PIR) problem, obtained by expanding the PIR message set (which consists of only independent messages) to also include functions of those messages. The capacity of private computation, $C$, is defined as the maximum …
The problem of Private Information Retrieval (PIR) from coded storage systems with colluding, byzantine, and unresponsive servers is considered. An explicit scheme using an $[n,k]$ Reed-Solomon storage code is designed, protecting against $t$-collusion and handling up to $b$ byzantine and $r$ unresponsive servers, when $n>k+t+2b+r-1$. This scheme achieves a PIR rate of $\frac{n-r-(k+2b+t-1)}{n-r}$. In the case where the capacity is known, namely when $k=1$, it is asymptotically capacity-achieving as the number of files grows. Lastly, the scheme is adapted to symmetric PIR.
We study two problems of private matrix multiplication, over a distributed computing system consisting of a master node, and multiple servers that collectively store a family of public matrices using Maximum-Distance-Separable (MDS) codes. In the first problem of Private and Secure Matrix Multiplication (PSMM) from colluding servers, the master intends to compute the product of its confidential matrix $\mathbf{A}$ with a target matrix stored on the servers, without revealing any information about $\mathbf{A}$ and the index of target matrix to some colluding servers. In the second problem of Fu…
In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by receiving quantum information from the servers. The QPIR problem has been treated by Song \emph{et al.} in the case of replicated servers, both without collusion and with all but one servers colluding. In this paper, the QPIR setting is extended to account for maximum distance separable (MDS) coded ser…
A private information retrieval (PIR) protocol guarantees that a user can privately retrieve files stored in a database without revealing any information about the identity of the requested file. Existing information-theoretic PIR protocols ensure perfect privacy, i.e., zero information leakage to the servers storing the database, but at the cost of high download. In this work, we present weakly-private information retrieval (WPIR) schemes that trade off perfect privacy to improve the download cost when the database is stored on a single server. We study the tradeoff between the download cost …
A new computational private information retrieval (PIR) scheme based on random linear codes is presented. A matrix of messages from a McEliece scheme is used to query the server with carefully chosen errors. The server responds with the sum of the scalar multiple of the rows of the query matrix and the files. The user recovers the desired file by erasure decoding the response. Contrary to code-based cryptographic systems, the scheme presented here enables to use truly random codes, not only codes disguised as such. Further, we show the relation to the so-called error subspace search problem an…
We consider the problem of private information retrieval (PIR) where a single user with private side information aims to retrieve multiple files from a library stored (uncoded) at a number of servers. We assume the side information at the user includes a subset of files stored privately (i.e., the server does not know the indices of these files). In addition, we require that the identity of the requests and side information at the user are not revealed to any of the servers. The problem involves finding the minimum load to be transmitted from the servers to the user such that the requested fil…
In this paper, we study the problem of \emph{private and secure distributed matrix multiplication (PSDMM)}, where a user having a private matrix $A$ and $N$ non-colluding servers sharing a library of $L$ ($L>1$) matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$, for which the user wishes to compute $AB^{(θ)}$ for some $θ\in [0, L)$ without revealing any information of the matrix $A$ to the servers, and keeping the index $θ$ private to the servers. Previous work is limited to the case that the shared library (\textit{i.e.,} the matrices $B^{(0)}, B^{(1)},\ldots,B^{(L-1)}$) is stored across the serve…
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 $(1+R+R^2+\cdots+R^{M-1})$…
This paper introduces a generalization of the Private Information Retrieval with Side Information (PIR-SI) problem called Popularity-Aware PIR-SI (PA-PIR-SI). The PA-PIR-SI problem includes one or more remote servers storing copies of a dataset of $K$ messages, and a user who knows $M$ out of $K$ messages -- the identities of which are unknown to the server -- as a prior side information, and wishes to retrieve one of the remaining $K-M$ messages. The goal of the user is to minimize the amount of information they must download from the server while revealing no information about the identity o…
This paper considers the problem of single-server single-message private information retrieval with coded side information (PIR-CSI). In this problem, there is a server storing a database, and a user which knows a linear combination of a subset of messages in the database as a side information. The number of messages contributing to the side information is known to the server, but the indices and the coefficients of these messages are unknown to the server. The user wishes to download a message from the server privately, i.e., without revealing which message it is requesting, while minimizing …
The problem of $X$-secure $T$-private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of $K$ independent messages that are distributed over $N$ servers according to an MDS code. It is guaranteed that any group of up to $X$ colluding servers learn nothing about the messages and that any group of up to $T$ colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on \emph{cross-subspace alignment} and a successive decoding with int…
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 …
A private information retrieval (PIR) scheme on coded storage systems with colluding, byzantine, and non-responsive servers is presented. Furthermore, the scheme can also be used for symmetric PIR in the same setting. An explicit scheme using an $[n,k]$ generalized Reed-Solomon storage code is designed, protecting against $t$-collusion and handling up to $b$ byzantine and $r$ non-responsive servers, when $n\geq n'= (ν+1) k+t+2b+r-1$, for some integer $ν\geq 1$. This scheme achieves a PIR rate of $1-\frac{k+2b+t+r-1}{n'}$. In the case where the capacity is known, namely when $k=1$, it is asympt…
This paper introduces the problem of Private Linear Transformation (PLT) which generalizes the problems of private information retrieval and private linear computation. The PLT problem includes one or more remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ independent linear combinations of a $D$-subset of messages. The objective of the user is to perform the computation by downloading minimum possible amount of information from the server(s), while protecting the identities of the $D$ messages required for the computation. In this work, we focus on…
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…
Quantum private information retrieval (QPIR) is a protocol in which a user retrieves one of multiple classical files by downloading quantum systems from non-communicating $\mathsf{n}$ servers each of which contains a copy of all files, while the identity of the retrieved file is unknown to each server. Symmetric QPIR (QSPIR) is QPIR in which the user only obtains the queried file but no other information of the other files. In this paper, we consider the $(\mathsf{n} - 1)$-private QSPIR in which the identity of the retrieved file is secret even if any $\mathsf{n} - 1$ servers collude, and deri…
We study the problem of single-server multi-message private information retrieval with side information. One user wants to recover $N$ out of $K$ independent messages which are stored at a single server. The user initially possesses a subset of $M$ messages as side information. The goal of the user is to download the $N$ demand messages while not leaking any information about the indices of these messages to the server. In this paper, we characterize the minimum number of required transmissions. We also present the optimal linear coding scheme which enables the user to download the demand mess…
We study the role of coded side information in single-server Private Information Retrieval (PIR). An instance of the single-server PIR problem includes a server that stores a database of $K$ independently and uniformly distributed messages, and a user who wants to retrieve one of these messages from the server. We consider settings in which the user initially has access to a coded side information which includes a linear combination of a subset of $M$ messages in the database. We assume that the identities of the $M$ messages that form the support set of the coded side information as well as t…
We introduce the problem of Private Linear Transformation (PLT). This problem includes a single (or multiple) remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ linear combinations of a $D$-subset of these messages by downloading the minimum amount of information from the server(s) while protecting the privacy of the entire set of $D$ messages. This problem generalizes the Private Information Retrieval and Private Linear Computation problems. In this work, we focus on the single-server case. For the setting in which the coefficient matrix of the req…
We study the problem of intermittent private information retrieval with multiple servers, in which a user consecutively requests one of K messages from N replicated databases such that part of requests need to be protected while others do not need privacy. Motivated by the location privacy application, the correlation between requests is modeled by a Markov chain. We propose an intermittent private information retrieval scheme that concatenates an obfuscation scheme and a private information retrieval scheme for the time period when privacy is not needed, to prevent leakage incurred by the cor…
The problem of symmetric private information retrieval (SPIR) from a coded database which is distributively stored among colluding servers is studied. Specifically, the database comprises $K$ files, which are stored among $N$ servers using an $(N,M)$-MDS storage code. A user wants to retrieve one file from the database by communicating with the $N$ servers, without revealing the identity of the desired file to any server. Furthermore, the user shall learn nothing about the other $K-1$ files in the database. In the $T$-colluding SPIR problem (hence called TSPIR), any $T$ out of $N$ servers may …
An information theoretic approach to security and privacy called Secure And Private Information Retrieval (SAPIR) is introduced. SAPIR is applied to distributed data storage systems. In this approach, random combinations of all contents are stored across the network. Our coding approach is based on Random Linear Fountain (RLF) codes. To retrieve a content, a group of servers collaborate with each other to form a Reconstruction Group (RG). SAPIR achieves asymptotic perfect secrecy if at least one of the servers within an RG is not compromised. Further, a Private Information Retrieval (PIR) sche…
We consider a multi-user variant of the private information retrieval problem described as follows. Suppose there are $D$ users, each of which wants to privately retrieve a distinct message from a server with the help of a trusted agent. We assume that the agent has a random subset of $M$ messages that is not known to the server. The goal of the agent is to collectively retrieve the users' requests from the server. For protecting the privacy of users, we introduce the notion of individual-privacy -- the agent is required to protect the privacy only for each individual user (but may leak some c…
The problem of Multi-user Blind $X$-secure $T$-colluding Symmetric Private Information Retrieval from Maximum Distance Separable (MDS) coded storage system with $B$ Byzantine and $U$ unresponsive servers (U-B-MDS-MB-XTSPIR) is studied in this paper. Specifically, a database consisting of multiple files, each labeled by $M$ indices, is stored at the distributed system with $N$ servers according to $(N,K+X)$ MDS codes over $\mathbb{F}_q$ such that any group of up to $X$ colluding servers learn nothing about the data files. There are $M$ users, in which each user $m,m=1,\ldots,M$ privately select…
In the classical private information retrieval (PIR) setup, a user wants to retrieve a file from a database or a distributed storage system (DSS) without revealing the file identity to the servers holding the data. In the quantum PIR (QPIR) setting, a user privately retrieves a classical file by receiving quantum information from the servers. The QPIR problem has been treated by Song et al. in the case of replicated servers, both with and without collusion. QPIR over $[n,k]$ maximum distance separable (MDS) coded servers was recently considered by Allaix et al., but the collusion was essential…
We study the capacity of quantum private information retrieval (QPIR) with multiple servers. In the QPIR problem with multiple servers, a user retrieves a classical file by downloading quantum systems from multiple servers each of which contains the copy of a classical file set while the identity of the downloaded file is not leaked to each server. The QPIR capacity is defined as the maximum rate of the file size over the whole dimension of the downloaded quantum systems. When the servers are assumed to share prior entanglement, we prove that the QPIR capacity with multiple servers is 1 regard…
Quantum private information retrieval (QPIR) is a protocol in which a user retrieves one of multiple files from $\mathsf{n}$ non-communicating servers by downloading quantum systems without revealing which file is retrieved. As variants of QPIR with stronger security requirements, symmetric QPIR is a protocol in which no other files than the target file are leaked to the user, and $\mathsf{t}$-private QPIR is a protocol in which the identity of the target file is kept secret even if at most $\mathsf{t}$ servers may collude to reveal the identity. The QPIR capacity is the maximum ratio of the f…
Motivated by recent interest in federated submodel learning, this work explores the fundamental problem of privately reading from and writing to a database comprised of $K$ files (submodels) that are stored across $N$ distributed servers according to an $X$-secure threshold secret sharing scheme. One after another, various users wish to retrieve their desired file, locally process the information and then update the file in the distributed database while keeping the identity of their desired file private from any set of up to $T$ colluding servers. The availability of servers changes over time…
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) protocols allow a user to retrieve entries of a database without revealing the index of the desired item. Information-theoretical privacy can be achieved by the use of several servers and specific retrieval algorithms. Most of known PIR protocols focus on decreasing the number of bits exchanged between the client and the server(s) during the retrieval process. On another side, Fazeli et. al. introduced so-called PIR codes in order to reduce the storage overhead on the servers. However, only a few works address the issue of the computation complexity of the s…
Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers. In this paper it is shown that Lagrange encoding, a powerful technique for encoding Reed-Solomon codes, enables private computation in many cases of interest. In particular, we present a scheme that enables private computation of polynomials of any degree on Lagrange encoded data, while being robust to Byzantine and straggling servers, and to servers colluding to attempt to deduce the ide…
This work investigates a system where each user aims to retrieve a scalar linear function of the files of a library, which are Maximum Distance Separable coded and stored at multiple distributed servers. The system needs to guarantee robust decoding in the sense that each user must decode its demanded function with signals received from any subset of servers whose cardinality exceeds a threshold. In addition, (a) the content of the library must be kept secure from a wiretapper who obtains all the signals from the servers;(b) any subset of users together can not obtain any information about the…
In this work, two practical concepts related to private information retrieval (PIR) are introduced and coined full support-rank PIR and strongly linear PIR. Being of full support-rank is a technical, yet natural condition required to prove a converse result for a capacity expression and satisfied by almost all currently known capacity-achieving schemes, while strong linearity is a practical requirement enabling implementation over small finite fields with low subpacketization degree. Then, the capacity of MDS-coded, linear, full support-rank PIR in the presence of colluding servers is derived,…
This paper considers the single-server Private Linear Transformation (PLT) problem with individual privacy guarantees. In this problem, there is a user that wishes to obtain $L$ independent linear combinations of a $D$-subset of messages belonging to a dataset of $K$ messages stored on a single server. The goal is to minimize the download cost while keeping the identity of each message required for the computation individually private. The individual privacy requirement ensures that the identity of each individual message required for the computation is kept private. This is in contrast to the…
In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size $t$ colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scen…
In private information delivery (PID) problem, there are $K$ messages stored across $N$ servers, each capable of storing $M$ messages and a user. Servers want to convey one of the $K$ messages to the user without revealing the identity (index) of the message conveyed. The capacity of PID problem is defined as maximum number of bits of the desired message that can be conveyed privately, per bit of total communication, to the user. For the restricted case of replicated systems, where coded messages or splitting one message into several servers is not allowed, the capacity of PID has been charact…
In a distributed storage system, private information retrieval (PIR) guarantees that a user retrieves one file from the system without revealing any information about the identity of its interested file to any individual server. In this paper, we investigate $(N,K,M)$ coded sever model of PIR, where each of $M$ files is distributed to the $N$ servers in the form of $(N,K)$ maximum distance separable (MDS) code for some $N>K$ and $M>1$. As a result, we propose a new capacity-achieving $(N,K,M)$ coded linear PIR scheme such that it can be implemented with file length $\frac{K(N-K)}{\gcd(N,K)}$, …
We study an information theoretic privacy mechanism design problem for two scenarios where the private data is either observable or hidden. In each scenario, we first consider bounded mutual information as privacy leakage criterion, then we use two different per-letter privacy constraints. In the first scenario, an agent observes useful data $Y$ that is correlated with private data $X$, and wishes to disclose the useful information to a user. A privacy mechanism is designed to generate disclosed data $U$ which maximizes the revealed information about $Y$ while satisfying a bounded privacy leak…
In many practical settings, the user needs to retrieve information from a server in a periodic manner, over multiple rounds of communication. In this paper, we discuss the setting in which this information needs to be retrieved privately, such that the identity of all the information retrieved until the current round is protected. This setting can occur in practical situations in which the user needs to retrieve items from the server or a periodic basis, such that the privacy needs to be guaranteed for all the items been retrieved until the current round. We refer to this setting as an \emph{o…
It was recently shown by Fazeli et al. that the storage overhead of a traditional $t$-server private information retrieval (PIR) protocol can be significantly reduced using the concept of a $t$-server PIR code. In this work, we show that a family of $t$-server PIR codes (with increasing dimensions and blocklengths) can be constructed from an existing $t$-server PIR code through lengthening by a single information symbol and code extension by at most $\bigl\lceil t/2\bigr\rceil$ code symbols. Furthermore, by extending a code construction notion from Steiner systems by Fazeli et al., we obtain a…
We consider a decentralized detection network whose aim is to infer a public hypothesis of interest. However, the raw sensor observations also allow the fusion center to infer private hypotheses that we wish to protect. We consider the case where there are an uncountable number of private hypotheses belonging to an uncertainty set, and develop local privacy mappings at every sensor so that the sanitized sensor information minimizes the Bayes error of detecting the public hypothesis at the fusion center, while achieving information privacy for all private hypotheses. We introduce the concept of…
In this paper, we study the multi-server setting of the \emph{Private Information Retrieval with Coded Side Information (PIR-CSI)} problem. In this problem, there are $K$ messages replicated across $N$ servers, and there is a user who wishes to download one message from the servers without revealing any information to any server about the identity of the requested message. The user has a side information which is a linear combination of a subset of $M$ messages in the database. The parameter $M$ is known to all servers in advance, whereas the indices and the coefficients of the messages in the…
