In this work, we consider private monomial computation (PMC) for replicated noncolluding databases. In PMC, a user wishes to privately retrieve an arbitrary multivariate monomial from a candidate set of monomials in $f$ messages over a finite field $\mathbb F_q$, where $q=p^k$ is a power of a prime $p$ and $k \ge 1$, replicated over $n$ databases. We derive the PMC capacity under a technical condition on $p$ and for asymptotically large $q$. The condition on $p$ is satisfied, e.g., for large enough $p$. Also, we present a novel PMC scheme for arbitrary $q$ that is capacity-achieving in the asymptotic case above. Moreover, we present formulas for the entropy of a multivariate monomial and for a set of monomials in uniformly distributed random variables over a finite field, which are used in the derivation of the capacity expression.
In this paper we study recent reaction attacks against QC-LDPC and QC-MDPC code-based cryptosystems, which allow an opponent to recover the private parity-check matrix through its distance spectrum by observing a sufficiently high number of decryption failures. We consider a special class of codes, known as monomial codes, to form private keys with the desirable property of having a unique and complete distance spectrum. We verify that for these codes the problem of recovering the secret key from the distance spectrum is equivalent to that of finding cliques in a graph, and use this equivalenc…
We study the private information retrieval (PIR) problem under arbitrary collusion pattern for replicated databases. We find its capacity, which is the same as the capacity of the original PIR problem with the number of databases $N$ replaced by a number $S^*$. The number $S^*$ is the optimal solution to a linear programming problem that is a function of the collusion pattern. Hence, the collusion pattern affects the capacity of the PIR problem only through the number $S^*$.
We consider a model for secrecy generation, with three terminals, by means of public interterminal communication, and examine the problem of characterizing all the rates at which all three terminals can generate a ``secret key,'' and -- simultaneously -- two designated terminals can generate a ``private key'' which is effectively concealed from the remaining terminal; both keys are also concealed from an eavesdropper that observes the public communication. Inner and outer bounds for the ``secret key--private key capacity region'' are derived. Under a certain special condition, these bounds coi…
We develop a new method for showing the optimality of the Gaussian distribution in multiterminal information theory problems. As an application of this method we show that Marton's inner bound achieves the capacity of the vector Gaussian broadcast channels with common message.
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 …
In this work, we investigate the capacity of private information retrieval (PIR) from $N$ replicated databases, where a subset of the databases are untrustworthy (byzantine) in their answers to the query of the user. We allow for multi-round queries and demonstrate that the identities of the byzantine databases can be determined with a small additional download cost. As a result, the capacity of the multi-round PIR with byzantine databases (BPIR) reaches that of the robust PIR problem when the number of byzantine databases is less than the number of trustworthy databases.
We study the symmetric private information retrieval (SPIR) problem under arbitrary collusion and eavesdropping patterns for replicated databases. We find its capacity, which is the same as the capacity of the original SPIR problem with the number of databases $N$ replaced by a number $F^*$. The number $F^*$ is the optimal solution to a linear programming problem that is a function of the joint pattern, which is the union of the collusion and eavesdropping pattern. This is the first result that shows how two arbitrary patterns collectively affect the capacity of the PIR problem. We draw the co…
The capacity of classical channels is convex. This is not the case for the quantum capacity of a channel: the capacity of a mixture of different quantum channels exceeds the mixture of the individual capacities and thus is non-convex. Here we show that this effect goes beyond the quantum capacity and holds for the private and classical environment-assisted capacities of quantum channels.
In multi user Gaussian relay networks, it is desirable to transmit private information to each user as well as common information to all of them. However, the capacity region of such networks with both kinds of information is not easy to characterize. The prior art used simple linear deterministic models in order to approximate the capacities of these Gaussian networks. This paper discusses the capacity region of the deterministic Y-channel with private and common messages. In this channel, each user aims at delivering two private messages to the other two users in addition to a common message…
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,…
The problem of simultaneously generating a secret key (SK) and private key (PK) pair among three terminals via public discussion is investigated, in which each terminal observes a component of correlated sources. All three terminals are required to generate a common secret key concealed from an eavesdropper that has access to public discussion, while two designated terminals are required to generate an extra private key concealed from both the eavesdropper and the remaining terminal. An outer bound on the SK-PK capacity region was established in [1], and was shown to be achievable for one case…
$X$-secure and $T$-private information retrieval (XSTPIR) is a form of private information retrieval where data security is guaranteed against collusion among up to $X$ servers and the user's privacy is guaranteed against collusion among up to $T$ servers. The capacity of XSTPIR is characterized for arbitrary number of servers $N$, and arbitrary security and privacy thresholds $X$ and $T$, in the limit as the number of messages $K\rightarrow\infty$. Capacity is also characterized for any number of messages if either $N=3, X=T=1$ or if $N\leq X+T$. Insights are drawn from these results, about a…
We study the capacity region of a deterministic 4-node network, where 3 nodes can only communicate via the fourth one. However, the fourth node is not merely a relay since it can exchange private messages with all other nodes. This situation resembles the case where a base station relays messages between users and delivers messages between the backbone system and the users. We assume an asymmetric scenario where the channel between any two nodes is not reciprocal. First, an upper bound on the capacity region is obtained based on the notion of single sided genie. Subsequently, we construct an a…
For a partially degradable (PD) channel, the channel output state can be used to simulate the degraded environment state. The quantum capacity of a PD channel has been proven to be additive. Here, we show that the private classical capacity of arbitrary dimensional PD channels is equal to the quantum capacity of the channel and also single-letterizes. We prove that higher rates of private classical communication can be achieved over a PD channel in comparison to standard degradable channels.
We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of $f$ messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of …
Private communication via zero-private-capacity quantum channels
QudeLeap Research（量子跃迁研究）；Thrust of Artificial Intelligence, Information Hub,The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州）信息枢纽人工智能组）
We consider the problem of $T$-Private Information Retrieval with private side information (TPIR-PSI). In this problem, $N$ replicated databases store $K$ independent messages, and a user, equipped with a local cache that holds $M$ messages as side information, wishes to retrieve one of the other $K-M$ messages. The desired message index and the side information must remain jointly private even if any $T$ of the $N$ databases collude. We show that the capacity of TPIR-PSI is $\left(1+\frac{T}{N}+\cdots+\left(\frac{T}{N}\right)^{K-M-1}\right)^{-1}$. As a special case obtained by setting $T=1$, …
This paper studies the private key generation of a cooperative pairwise-independent network (PIN) with M+2 terminals (Alice, Bob and M relays), M >= 2. In this PIN, the correlated sources observed by every pair of terminals are independent of those sources observed by any other pair of terminal. All the terminals can communicate with each other over a public channel which is also observed by Eve noiselessly. The objective is to generate a private key between Alice and Bob under the help of the M relays; such a private key needs to be protected not only from Eve but also from individual relays …
We consider a model with three terminals and examine the problem of characterizing the largest rates at which two pairs of terminals can simultaneously generate private keys, each of which is effectively concealed from the remaining terminal.
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 …
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 …
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…
This paper considers the problem of single-server Private Computation (PC) in the presence of Side Information (SI). In this problem, there is a server that stores $K$ i.i.d. messages, and a user who has a subset of $M$ uncoded messages or a coded linear combination of them as side information, where the identities of these messages are unknown to the server. The user wants to privately compute (via downloading information from the server) a linear combination of a subset of $D$ other messages, where the identities of these messages must be kept private individually or jointly. For each settin…
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 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 define a new phenomenon for communication over noisy quantum channels. The investigated solution is called polaractivation and based on quantum polar encoding. Polaractivation is a natural consequence of the channel polarization effect in quantum systems and makes possible to open the hidden capacity regions of a noisy quantum channel by using the idea of rate increment. While in case of a classical channel only the rate of classical communication can be increased, in case of a quantum channel the channel polarization and the rate improvement can be exploited to open unreachable capacity re…
The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. However, both quantities are given by the infinite regularization of respectively the coherent and the private information. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses. We prove this by showing that the coherent information is strictly larger than the private information of a smaller number of uses of the channel. This implies that even though the quantum capacity is uppe…
The private search problem is introduced, where a dataset comprised of $L$ i.i.d. records is replicated across $N$ non-colluding servers, each record takes values uniformly from an alphabet of size $K$, and a user wishes to search for all records that match a privately chosen value, without revealing any information about the chosen value to any individual server. The capacity of private search is the maximum number of bits of desired information that can be retrieved per bit of download. The asymptotic (large $K$) capacity of private search is shown to be $1-1/N$, even as the scope of private…
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…
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 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 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…
In the private information retrieval (PIR) problem a user wishes to retrieve, as efficiently as possible, one out of $K$ messages from $N$ non-communicating databases (each holds all $K$ messages) while revealing nothing about the identity of the desired message index to any individual database. The information theoretic capacity of PIR is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. For $K$ messages and $N$ databases, we show that the PIR capacity is $(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. A remarkable feature of the capac…
We investigate the problem of information theoretically secure communication in a line network with erasure channels and state feedback. We consider a spectrum of cases for the private randomness that intermediate nodes can generate, ranging from having intermediate nodes generate unlimited private randomness, to having intermediate nodes generate no private randomness, and all cases in between. We characterize the secret message capacity when either only one of the channels is eavesdropped or all of the channels are eavesdropped, and we develop polynomial time algorithms that achieve these ca…
In the vein of the recent "pretty strong" converse for the quantum and private capacity of degradable quantum channels [Morgan/Winter, IEEE Trans. Inf. Theory 60(1):317-333, 2014], we use the same techniques, in particular the calculus of min-entropies, to show a pretty strong converse for the private capacity of degraded classical-quantum-quantum (cqq-)wiretap channels, which generalize Wyner's model of the degraded classical wiretap channel. While the result is not completely tight, leaving some gap between the region of error and privacy parameters for which the converse bound holds, and a …
The K-user discrete memoryless (DM) broadcast channel (BC) with two nested multicast messages is studied in which one common message is to be multicast to all receivers and the second private message to a subset of receivers. The receivers that must decode both messages are referred to as private receivers and the others that must decode only the common message as common receivers. For two nested multicast messages, we establish the capacity region for several classes of DM BCs characterized by the respective associated sets of pair-wise relationships between and among the common and private r…
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 …
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…
Two new classes of quantum channels, which we call more capable and less noisy, are introduced. The more capable class consists of channels such that the quantum capacities of the complementary channels to the environments are zero. The less noisy class consists of channels such that the private capacities of the complementary channels to the environment are zero. For the more capable class, it is clarified that the private capacity and quantum capacity coincide. For the less noisy class, it is clarified that the private capacity and quantum capacity can be single letter characterized.
We characterize the capacity region of the symmetric injective K-user Deterministic Interference Channel (DIC) for all channel parameters. The achievable rate region is derived by first projecting the achievable rate region of Han-Kobayashi (HK) scheme, which is in terms of common and private rates for each user, along the direction of aggregate rates for each user (i.e., the sum of common and private rates). We then show that the projected region is characterized by only the projection of those facets in the HK region for which the coefficient of common rate and private rate are the same for …
We propose capacity-achieving schemes for private information retrieval (PIR) from uncoded databases (DBs) with both homogeneous and heterogeneous storage constraints. In the PIR setting, a user queries a set of DBs to privately download a message, where privacy implies that no one DB can infer which message the user desires. In general, a PIR scheme is comprised of storage placement and delivery designs. Previous works have derived the capacity, or infimum download cost, of PIR with uncoded storage placement and also sufficient conditions of a storage placement design to meet capacity. Howeve…
We study Private Information Retrieval with Side Information (PIR-SI) in the single-server multi-message setting. In this setting, a user wants to download $D$ messages from a database of $K\geq D$ messages, stored on a single server, without revealing any information about the identities of the demanded messages to the server. The goal of the user is to achieve information-theoretic privacy by leveraging the side information about the database. The side information consists of a random subset of $M$ messages in the database which could have been obtained in advance from other users or from pr…
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 …
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 $S$ possesses side information in the form of uncoded portions of the messages that are unknown to the databases. The identities of these side information messages need to be kept private from the databases, i.e., we consider PIR with private side information (PSI). We characterize the optimal normalized download cost for this PIR-PSI problem under the storage constraint $S$ as $D^*=1+\frac{1}…
Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other $K-1$ messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired infor…
Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of $K$ messages from $N$ non-communicating replicated databases (each holds all $K$ messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. $T$-private PIR is a generalization of PIR to include the requirement that even if any $T$ …
We show that it is possible for the so-called weak locking capacity of a quantum channel [Guha et al., PRX 4:011016, 2014] to be much larger than its private capacity. Both reflect different ways of capturing the notion of reliable communication via a quantum system while leaking almost no information to an eavesdropper; the difference is that the latter imposes an intrinsically quantum security criterion whereas the former requires only a weaker, classical condition. The channels for which this separation is most straightforward to establish are the complementary channels of classical-quantum…
We consider the problem of private information retrieval (PIR) with colluding servers and eavesdroppers (abbreviated as ETPIR). The ETPIR problem is comprised of $K$ messages, $N$ servers where each server stores all $K$ messages, a user who wants to retrieve one of the $K$ messages without revealing the desired message index to any set of $T$ colluding servers, and an eavesdropper who can listen to the queries and answers of any $E$ servers but is prevented from learning any information about the messages. The information theoretic capacity of ETPIR is defined to be the maximum number of desi…
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…
The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary $T$-privacy and $X$-security constraints, where the privacy of the user must be protected against any set of up to $T$ colluding servers and the security of the stored data must be protected against any set of up to $X$ colluding servers. A general achievable scheme for arbitr…
We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of $N$ non-colluding databases, each storing a coded version of $M$ messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is $C…
The capacity has recently been characterized for the private information retrieval (PIR) problem as well as several of its variants. In every case it is assumed that all the queries are generated by the user simultaneously. Here we consider multiround PIR, where the queries in each round are allowed to depend on the answers received in previous rounds. We show that the capacity of multiround PIR is the same as the capacity of single-round PIR (the result is generalized to also include $T$-privacy constraints). Combined with previous results, this shows that there is no capacity advantage from …
