Private Information Retrieval with Private Noisy Side Information

Consider Private Information Retrieval (PIR), where a client wants to retrieve one file out of $K$ files that are replicated in $N$ different servers and the client selection must remain private when up to $T$ servers may collude. Additionally, suppose that the client has noisy side information about each of the $K$ files, and the side information about a specific file is obtained by passing this file through one of $D$ possible discrete memoryless test channels, where $D\le K$. While the statistics of the test channels are known by the client and by all the servers, the specific mapping $\boldsymbol{\mathcal{M}}$ between the files and the test channels is unknown to the servers. We study this problem under two different privacy metrics. Under the first privacy metric, the client wants to preserve the privacy of its desired file selection and the mapping $\boldsymbol{\mathcal{M}}$. Under the second privacy metric, the client wants to preserve the privacy of its desired file and the mapping $\boldsymbol{\mathcal{M}}$, but is willing to reveal the index of the test channel that is associated to its desired file. For both of these two privacy metrics, we derive the optimal normalized download cost. Our problem setup generalizes PIR with colluding servers, PIR with private noiseless side information, and PIR with private side information under storage constraints

Single-Server Multi-Message Private Information Retrieval with Side Information

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 messages and preserves the privacy of their indices. Moreover, we show that the trivial MDS coding scheme with $K-M$ transmissions is optimal if $N>M$ or $N^2+N \ge K-M$. This means if one wishes to privately download more than the square-root of the number of files in the database, then one must effectively download the full database (minus the side information), irrespective of the amount of side information one has available.Comment: 12 pages, submitted to the 56th Allerton conferenc

Single-Server Single-Message Online Private Information Retrieval with Side Information

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{online private information retrieval} as the user does not know the identities of the future items that need to be retrieved from the server. Following the previous line of work by Kadhe \emph{et al.}~we assume that the user knows a random subset of $M$ messages in the database as a side information which are unknown to the server. Focusing on scalar-linear settings, we characterize the \emph{per-round capacity}, i.e., the maximum achievable download rate at each round, and present a coding scheme that achieves this capacity. The key idea of our scheme is to utilize the data downloaded during the current round as a side information for the subsequent rounds. We show for the setting with $K$ messages stored at the server, the per-round capacity of the scalar-linear setting is $C_1= ({M+1})/{K}$ for round $i=1$ and ${C_i= {(2^{i-1}(M+1))}/{KM}}$ for round $i\geq2$, provided that ${K}/({M+1})$ is a power of $2$.Comment: 7 pages; This work is a long version of an article submitted to IEEE for possible publicatio

Private Information Retrieval with Side Information

The objective of the classical Private Information Retrieval (PIR) problem is to enable a user to download a message from a database that is replicated across a collection of non-colluding servers without revealing the identity of the demanded message to the servers. In the classical PIR problem the user has no prior information about the content of messages in the database. It is easy to verify in the special case of the PIR problem when there is only one server in the system, the user must download all messages from the database in order keep information about the message they want private. In a real environment the user may have other sources to download and obtain messages from such as trusted peer-to-peer communication. In this way, the user has the potential to obtain some of the messages that are contained in the database to use as side information in a PIR scheme with the servers. Accordingly, we introduce the Private Information Retrieval with Side Information (PIR-SI) problem that focuses on settings in which the user has side information about some messages in the database. To capture the different levels of privacy a user may want to enforce in PIR-SI schemes two metrics of privacy, W-privacy and (W,S)-privacy, are introduced. W-privacy only protects information about the identity of the message that the user wants and is most similar to the measure of privacy in the original PIR problem. (W, S)-privacy protects the identity of the wanted message as well as the identities of the messages they have as side information and is a stronger sense of privacy than W-privacy. When enforcing either measure of privacy the user no longer has to download all the messages in the database, even if there is only one server in the system; side information reduces the amount of data that one has to download in a PIR scheme. The first case of the PIR-SI problem that we consider is when the user has M messages for side information and wants a different message from the database of K messages. When there is only one server in the system, we show that the optimal download rate for a W-private scheme is k^-1/M+1 and the optimal download rate for a (W, S)-private scheme is 1/K-M. When there is more than one server in the system a W-private scheme is presented that has a larger rate than the classical PIR scheme, but its optimality is not shown. The second case of the PIR-SI problem that is considered is when the user has M messages as side information, the user wants D > 1 distinct messages from the database, and there is only one server in the system. In the case when M = 1 a (W, S)-private optimal scheme is presented and shown to be optimal. In the case when M ≥ D and D = 2 a W-private scheme that can increase the rate from the (W, S)-private scheme with the same parameters is presented. This scheme’s optimality reminds an open problem. We highlight the difficulty of finding an optimal scheme and determining the capacity of the multi-message PIR-SI problem