Capacity Of Private Information Retrieval Research Articles

New private information retrieval codes for imperfect privacy

In this paper, we introduce new codes for the private information retrieval (PIR). The conventional Tian-Sun-Chen (TSC) scheme achieves PIR capacity under the perfect privacy conditions with the minimum message length. Based on the TSC scheme, weakly PIR (WPIR) schemes have been developed to benefit download costs by relaxing privacy. We show that the proposed scheme has lower download cost than the TSC scheme in the weak privacy setting. While our scheme is not capacity-achieving under perfect privacy, it has smaller message sizes compared with the TSC scheme.

Deceptive Information Retrieval.

We introduce the problem of deceptive information retrieval (DIR), in which a user wishes to download a required file out of multiple independent files stored in a system of databases while deceiving the databases by making the databases' predictions on the user-required file index incorrect with high probability. Conceptually, DIR is an extension of private information retrieval (PIR). In PIR, a user downloads a required file without revealing its index to any of the databases. The metric of deception is defined as the probability of error of databases' prediction on the user-required file, minus the corresponding probability of error in PIR. The problem is defined on time-sensitive data that keep updating from time to time. In the proposed scheme, the user deceives the databases by sending real queries to download the required file at the time of the requirement and dummy queries at multiple distinct future time instances to manipulate the probabilities of sending each query for each file requirement, using which the databases' make the predictions on the user-required file index. The proposed DIR scheme is based on a capacity achieving probabilistic PIR scheme, and achieves rates lower than the PIR capacity due to the additional downloads made to deceive the databases. When the required level of deception is zero, the proposed scheme achieves the PIR capacity.

Bounds on the Capacity of Private Information Retrieval Over Graphs

In the private information retrieval (PIR) problem, a user wants to retrieve a file from a database without revealing any information about the desired file’s identity to the servers that store the database. In this paper, we study the PIR capacity of a graph-based replication system, in which each file is stored on two distinct servers according to an underlying graph. This paper aims to provide upper and lower bounds to the PIR capacity of graphs via various graph properties. In particular, we provide several upper bounds on the PIR capacity that apply to all graphs. We further improve the bounds for specific graph families (which turn out to be tight in certain cases) by utilizing the underlying graph structure. For the lower bounds, we establish optimal rate PIR schemes for star graphs via edge-coloring techniques. Lastly, we provide an improved PIR scheme for complete graphs, implying an improved general lower bound on all graphs’ PIR capacity.

Toward the Capacity of Private Information Retrieval From Coded and Colluding Servers

In this work, two practical concepts related to private information retrieval (PIR) are introduced and coined <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">full support-rank</i> PIR and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strongly linear</i> 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, as well as the capacity of symmetric, linear PIR with colluding, adversarial, and nonresponsive servers for the recently introduced concept of matched randomness. This positively settles the capacity conjectures stated by Freij-Hollanti <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> and Tajeddine <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in the presented cases. It is also shown that, further restricting to strongly-linear PIR schemes with deterministic linear interference cancellation, the so-called star product scheme proposed by Freij-Hollanti <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> is essentially optimal and induces no capacity loss.

The Capacity of Private Information Retrieval Under Arbitrary Collusion Patterns for Replicated Databases

We study the private information retrieval (PIR) problem under arbitrary collusion patterns for replicated databases. We find a general characterization of the PIR capacity, which is the same as the capacity of the original PIR problem with the number of databases N replaced by a number S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> . S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> is the optimal solution to a linear programming problem that is a function of the incidence matrix of the collusion pattern. Hence, the essence of any collusion pattern can be distilled into one number S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> . In the proposed achievable scheme, databases are non-uniformly queried according to the optimal solution of a linear programming problem based on the collusion pattern. It can be seen that the databases who collude more with others are queried less. In the converse proof, Shearer's lemma is applied, in place of Han's inequality, to a linear combination of inequalities, where each inequality corresponds to one colluding set in the collusion pattern. The weights of the linear combination come from the optimal solution of another linear programming problem based on the collusion pattern. Finally, by noting the interesting fact that the two seemingly different linear programming problems, one used in the achievability proof and the other used in the converse proof, are in fact dual problems, we characterize the capacity of the PIR problem under arbitrary collusion patterns.

Private Information Retrieval Schemes With Product-Matrix MBR Codes

A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. As of now, PIR schemes have been proposed for several kinds of storage systems, including replicated and MDS-coded systems. However, the problem of constructing PIR schemes on regenerating codes has been sparsely considered. A regenerating code is a storage code whose codewords are distributed among nodes, enabling efficient storage of files, as well as low-bandwidth retrieval of files and repair of nodes. Minimum-bandwidth regenerating (MBR) codes define a family of regenerating codes allowing a node repair with optimal bandwidth. Rashmi, Shah, and Kumar obtained a large family of MBR codes using the product-matrix (PM) construction. In this work, a new PIR scheme over PM-MBR codes is designed. The inherent redundancy of the PM structure is used to reduce the download communication complexity of the scheme. A lower bound on the PIR capacity of MBR-coded PIR schemes is derived, showing an interesting storage space vs. PIR rate trade-off compared to existing PIR schemes with the same reconstruction capability. The present scheme also outperforms a recent PM-MBR PIR construction of Dorkson and Ng.

The Capacity of Private Information Retrieval From Uncoded Storage Constrained Databases

Private information retrieval (PIR) allows a user to retrieve a desired message from a set of databases without revealing the identity of the desired message. The replicated database scenario, where $N$ databases store each of the $K$ messages was considered by Sun and Jafar, and the optimal download cost was characterized as $\left ({1+ \frac {1}{N}+ \frac {1}{N^{2}}+ \cdots + \frac {1}{N^{K-1}}}\right)$ . In this work, we consider the problem of PIR from uncoded storage constrained databases. Each database has a storage capacity of $\mu KL$ bits, where $L$ is the size of each message in bits, and $\mu \in [{1/N, 1}]$ is the normalized storage. The novel aspect of this work is to characterize the optimum download cost of PIR from uncoded storage constrained databases for any “normalized storage” value in the range $\mu \in [{1/N, 1}]$ . In particular, for any $(N,K)$ , we show that the optimal trade-off between normalized storage, $\mu $ , and the download cost, $D(\mu)$ , is a piece-wise linear function given by the lower convex hull of the $N$ pairs $\left ({\frac {t}{N}, \left ({1+ \frac {1}{t}+ \frac {1}{t^{2}}+ \cdots + \frac {1}{t^{K-1}}}\right)}\right)$ for $t=1,2,\ldots, N$ . To prove this result, we first present a storage constrained PIR scheme for any $(N,K)$ . Next, we obtain a general lower bound on the download cost for PIR, which is valid for any arbitrary storage architecture. The uncoded storage assumption is then applied which allows us to express the lower bound as a linear program (LP). Finally, we solve the LP to obtain tight lower bounds on the download cost for different regimes of storage, which match the proposed storage constrained PIR scheme.

Towards Practical Private Information Retrieval From MDS Array Codes

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 schemes from MSR coded servers either require large sub-packetization levels or are not capacity-achieving. In this paper, a PIR protocol from MDS array codes is proposed, subsuming PIR from MSR coded servers as a special case. Particularly, the case of non-colluding, honest-but-curious servers is considered. The retrieval rate of the new PIR protocol achieves the capacity of PIR from MDS/MSR coded servers. By choosing different MDS array codes, the new PIR protocol can have some advantages when compared with existing protocols, e.g., 1) small sub-packetization, 2) (near-) optimal repair bandwidth, 3) implementable over the binary field $\mathbf{F}_2$.

The Capacity of Private Information Retrieval From Heterogeneous Uncoded Caching Databases

We consider private information retrieval (PIR) of a single file out of K files from N non-colluding databases with heterogeneous storage constraints m = (m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , ⋯, m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ). The aim of this work is to jointly design the content placement phase and the information retrieval phase in order to minimize the download cost in the PIR phase. We characterize the optimal PIR download cost as a linear program. By analyzing the structure of the optimal solution of this linear program, we show that, surprisingly, the optimal download cost in our heterogeneous case matches its homogeneous counterpart where all databases have the same average storage constraint μ = 1/N Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> mn. N Thus, we show that there is no loss in the PIR capacity due to heterogeneity of storage spaces of the databases. We provide the optimum content placement explicitly for N = 3.

One-Shot PIR: Refinement and Lifting

We study a class of private information retrieval (PIR) methods that we call one-shot schemes. The intuition behind one-shot schemes is the following. The user’s query is regarded as a dot product of a query vector and the message vector (database) stored at multiple servers. Privacy, in an information theoretic sense, is then achieved by encrypting the query vector using a secure linear code, such as secret sharing. Several PIR schemes in the literature, in addition to novel ones constructed here, fall into this class. One-shot schemes provide an insightful link between PIR and data security against eavesdropping. However, their download rate is not optimal, i.e., they do not achieve the PIR capacity. Our main contribution is two transformations of one-shot schemes, which we call refining and lifting. We show that refining and lifting one-shot schemes gives capacity-achieving schemes for the cases when the PIR capacity is known. In the other cases, when the PIR capacity is still unknown, refining and lifting one-shot schemes gives, for most parameters, the best download rate so far.

The Capacity of Private Information Retrieval With Partially Known Private Side Information

We consider the problem of private information retrieval (PIR) of a single message out of $K$ messages from $N$ replicated and non-colluding databases where a cache-enabled user (retriever) of cache-size $M$ possesses side information in the form of full messages that are partially known to the databases. In this model, the user and the databases engage in a two-phase scheme, namely, the prefetching phase where the user acquires side information and the retrieval phase where the user downloads desired information. In the prefetching phase, the user receives $m_{n}$ full messages from the $n$ th database, under the cache memory size constraint $\sum _{n=1}^{N} m_{n} \leq M$ . In the retrieval phase, the user wishes to retrieve a message (which is not present in its memory) such that no individual database learns anything about the identity of the desired message. In addition, the identities of the side information messages that the user did not prefetch from a database must remain private against that database. Since the side information provided by each database in the prefetching phase is known by the providing database and the side information must be kept private against the remaining databases, we coin this model as partially known private side information . We characterize the capacity of the PIR with partially known private side information to be $C=\left ({1+\frac {1}{N}+\cdots +\frac {1}{N^{K-M-1}}}\right)^{-1}=\frac {1-\frac {1}{N}}{1-\left({\frac {1}{N}}\right)^{K-M}}$ . Interestingly, this result is the same if none of the databases knows any of the prefetched side information, i.e., when the side information is obtained externally, a problem posed by Kadhe et al. and settled by Chen-Wang-Jafar recently. Thus, our result implies that there is no loss in using the same databases for both prefetching and retrieval phases.

The Capacity of Private Information Retrieval from Decentralized Uncoded Caching Databases

We consider the private information retrieval (PIR) problem from decentralized uncoded caching databases. There are two phases in our problem setting, a caching phase, and a retrieval phase. In the caching phase, a data center containing all the K files, where each file is of size L bits, and several databases with storage size constraint μ K L bits exist in the system. Each database independently chooses μ K L bits out of the total K L bits from the data center to cache through the same probability distribution in a decentralized manner. In the retrieval phase, a user (retriever) accesses N databases in addition to the data center, and wishes to retrieve a desired file privately. We characterize the optimal normalized download cost to be D * = ∑ n = 1 N + 1 N n - 1 μ n - 1 ( 1 - μ ) N + 1 - n 1 + 1 n + ⋯ + 1 n K - 1 . We show that uniform and random caching scheme which is originally proposed for decentralized coded caching by Maddah-Ali and Niesen, along with Sun and Jafar retrieval scheme which is originally proposed for PIR from replicated databases surprisingly results in the lowest normalized download cost. This is the decentralized counterpart of the recent result of Attia, Kumar, and Tandon for the centralized case. The converse proof contains several ingredients such as interference lower bound, induction lemma, replacing queries and answering string random variables with the content of distributed databases, the nature of decentralized uncoded caching databases, and bit marginalization of joint caching distributions.

Capacity-Achieving Private Information Retrieval Codes With Optimal Message Size and Upload Cost

We propose a new capacity-achieving code for the private information retrieval (PIR) problem, and show that it has the minimum message size (being one less than the number of servers) and the minimum upload cost (being roughly linear in the number of messages) among a general class of capacity-achieving codes, and in particular, among all capacity-achieving linear codes. Different from existing code constructions, the proposed code is asymmetric, and this asymmetry appears to be the key factor leading to the optimal message size and the optimal upload cost. The converse results on the message size and the upload cost are obtained by an analysis of the information theoretic proof of the PIR capacity, from which a set of critical properties of any capacity-achieving code in the code class of interest is extracted. The symmetry structure of the PIR problem is then analyzed, which allows us to construct symmetric codes from asymmetric ones, yielding a meaningful bridge between the proposed code and existing ones in the literature.

Asymmetry Hurts: Private Information Retrieval Under Asymmetric Traffic Constraints

We consider the classical setting of private information retrieval (PIR) of a single message (file) out of $M$ messages from $N$ distributed databases under the new constraint of asymmetric traffic from databases. In this problem, the ratios between the traffic from the databases are constrained, i.e., the ratio of the length of the answer string that the user (retriever) receives from the $n$ th database to the total length of all answer strings from all databases is constrained to be $\tau _{n}$ . This may happen if the user’s access to the databases is restricted due to database availability, channel quality to the databases, and other factors. For this problem, for fixed $M$ , $N$ , we develop a general upper bound $\bar {C}({\boldsymbol {\tau }})$ , which generalizes the converse proof of Sun-Jafar, where database symmetry was inherently used. Our converse bound is a piece-wise affine function in the traffic ratio vector ${\boldsymbol {\tau }}=(\tau _{1}, \cdots , \tau _{N})$ . For the lower bound, we explicitly show the achievability of $\binom {M+N-1}{M}$ corner points. For the remaining traffic ratio vectors, we perform time-sharing between these corner points. The recursive structure of our achievability scheme is captured via a system of difference equations. The upper and lower bounds exactly match for $M=2$ and $M=3$ for any $N$ and any ${\boldsymbol {\tau }}$ . The results show strict loss of PIR capacity due to the asymmetric traffic constraints compared with the symmetric case of Sun-Jafar which implicitly uses $\tau _{n}=\frac {1}{N}$ for all $n$ .

Breaking the MDS-PIR Capacity Barrier via Joint Storage Coding

The capacity of private information retrieval (PIR) from databases coded using maximum distance separable (MDS) codes was previously characterized by Banawan and Ulukus, where it was assumed that the messages are encoded and stored separably in the databases. This assumption was also usually made 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 of if and when this capacity barrier can be broken through joint encoding and storing of the messages. Our main results are two classes of novel code constructions, which allow joint encoding, as well as the corresponding PIR protocols, which indeed outperformed the separate MDS-coded systems. Moreover, we show that a simple, but novel expansion technique allows us to generalize these two classes of codes, resulting in a wider range of the cases where this capacity barrier can be broken.

The Capacity of Private Computation

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 number of bits of the desired function that can be retrieved per bit of total download from all servers. We characterize the capacity of private computation, for $N$ servers and $K$ independent datasets that are replicated at each server, when the functions to be computed are arbitrary linear combinations of the datasets. Surprisingly, the capacity, $C=\left(1+1/N+\cdots+1/N^{K-1}\right)^{-1}$, matches the capacity of PIR with $N$ servers and $K$ messages. Thus, allowing arbitrary linear computations does not reduce the communication rate compared to pure dataset retrieval. The same insight is shown to hold even for arbitrary non-linear computations when the number of datasets $K\rightarrow\infty$.

A general private information retrieval scheme for MDS coded databases with colluding servers

The problem of private information retrieval (PIR) gets renewed attentions in recent years due to its information-theoretic reformulation and applications in distributed storage systems. Let M files be stored in a distributed storage system consisting of N servers, where each file is stored via an (N, K)-MDS code. A PIR scheme will allow a user to retrieve a specific file from the distributed storage system without revealing the identity of the file to any T colluding servers. PIR rate is defined as the number of symbols privately retrieved per one downloaded symbol and the supremum of all achievable rates is called the PIR capacity. The capacity has been solved for some degenerate cases, i.e. $$K=1$$ or $$T=1$$ . For the general case $$K\ge 2$$ and $$T\ge 2$$ , the exact PIR capacity remains unknown. In this paper we propose a general private information retrieval scheme for MDS coded databases with colluding servers achieving PIR rate $$(1+R+R^2+\cdots +R^{M-1})$$ , where $$R=1-\frac{{{N-T}\atopwithdelims ()K}}{{N\atopwithdelims ()K}}$$ . Our scheme captures the essence of the optimal schemes for degenerate cases. We also compare our scheme with some other known PIR schemes for non-degenerate cases. The advantages of our scheme include its independence of the property of the storage code and a better performance when the number of files M is small.

$t$-private information retrieval schemes using transitive codes

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, 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. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed–Muller codes serve as an important and explicit class of examples.

Achieving Maximum Distance Separable Private Information Retrieval Capacity With Linear Codes

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 rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed–Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al. . Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates.

The Capacity of Private Information Retrieval from Byzantine and Colluding Databases

We consider the problem of single-round private information retrieval (PIR) from N replicated databases. We consider the case when B databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of M messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the T-privacy constraint in this paper, where any T databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of correct symbols that can be retrieved privately (under the T-privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be C = (N -2B)/N·(1-T/(N-2B))/(1-(T/(N - 2B)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sup> ), if 2B + T <; N. This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing 2B databases from the system, with a penalty factor of (N - 2B)/N, which signifies that even though the number of databases needed for PIR is effectively N - 2B, the user still needs to access the entire N databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are missynchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the errors introduced by the Byzantine databases as opposed to erasures in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes.