arXiv subjects

Karim Banawan

Publications and source records attributed to Karim Banawan.

26 records · Page 2Linked to original sources

arXiv2018

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 \emph{asymmetric traffic} from databases. In this problem, the \emph{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 $τ_n$. This may happen if the user's access to the databases is restricted due database availability, channel quality to the databases, and other factors. For this problem, for fixed $M$, $N$, we develop a general upper bound $\bar{C}(\boldsymbolτ)$, 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 $\boldsymbolτ=(τ_1, \cdots, τ_N)$. 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 $\boldsymbolτ$. 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 $τ_n=\frac{1}{N}$ for all $n$.

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arXiv2017

Cache-Aided Private Information Retrieval with Partially Known Uncoded Prefetching: Fundamental Limits

We consider the problem of private information retrieval (PIR) from $N$ non-colluding and replicated databases, when the user is equipped with a cache that holds an uncoded fraction $r$ of the symbols from each of the $K$ stored messages in the databases. This model operates in a two-phase scheme, namely, the prefetching phase where the user acquires side information and the retrieval phase where the user privately downloads the desired message. In the prefetching phase, the user receives $\frac{r}{N}$ uncoded fraction of each message from the $n$th database. This side information is known only to the $n$th database and unknown to the remaining databases, i.e., the user possesses \emph{partially known} side information. We investigate the optimal normalized download cost $D^*(r)$ in the retrieval phase as a function of $K$, $N$, $r$. We develop lower and upper bounds for the optimal download cost. The bounds match in general for the cases of very low caching ratio ($r \leq \frac{1}{N^{K-1}}$) and very high caching ratio ($r \geq \frac{K-2}{N^2-3N+KN}$). We fully characterize the optimal download cost caching ratio tradeoff for $K=3$. For general $K$, $N$, and $r$, we show that the largest gap between the achievability and the converse bounds is $\frac{5}{32}$.

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arXiv2017

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 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 \textit{partially known private side information}. We characterize the capacity of the PIR with partially known private side information to be $C=\left(1+\frac{1}{N}+\cdots+\frac{1}{N^{K-M-1}}\right)^{-1}=\frac{1-\frac{1}{N}}{1-(\frac{1}{N})^{K-M}}$. 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.

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arXiv2017

Fundamental Limits of Cache-Aided Private Information Retrieval with Unknown and Uncoded Prefetching

We consider the problem of private information retrieval (PIR) from $N$ non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction $r$ from each of the $K$ stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate $D^*(r)$ the optimal download cost normalized with the message size as a function of $K$, $N$, $r$. For a fixed $K$, $N$, we develop an inner bound (converse bound) for the $D^*(r)$ curve. The inner bound is a piece-wise linear function in $r$ that consists of $K$ line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve $K-1$ non-degenerate corner points. These corner points differ in the number of cached bits that are used to generate one side information equation. We obtain an outer bound (achievability) for any caching ratio by memory-sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in $r$ that consists of $K$ line segments. The inner and the outer bounds match in general for the cases of very low caching ratio ($r \leq \frac{1}{1+N+N^2+\cdots+N^{K-1}}$) and very high caching ratio ($r \geq \frac{K-2}{(N+1)K+N^2-2N-2}$). As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for $K=3$. For general $K$, $N$, and $r$, we show that the largest gap between the achievability and the converse bounds is $\frac{1}{6}$. Our results show that the download cost can be reduced beyond memory-sharing if the databases are unaware of the cached content.

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arXiv2017

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 \emph{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 $C=\frac{N-2B}{N}\cdot\frac{1-\frac{T}{N-2B}}{1-(\frac{T}{N-2B})^M}$, 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 $\frac{N-2B}{N}$, 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 mis-synchronized, 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 \emph{errors} introduced by the Byzantine databases as opposed to \emph{erasures} in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes.

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arXiv2017

Multi-Message Private Information Retrieval: Capacity Results and Near-Optimal Schemes

We consider the problem of multi-message private information retrieval (MPIR) from $N$ non-communicating replicated databases. In MPIR, the user is interested in retrieving $P$ messages out of $M$ stored messages without leaking the identity of the retrieved messages. The information-theoretic sum capacity of MPIR $C_s^P$ is the maximum number of desired message symbols that can be retrieved privately per downloaded symbol. For the case $P \geq \frac{M}{2}$, we determine the exact sum capacity of MPIR as $C_s^P=\frac{1}{1+\frac{M-P}{PN}}$. The achievable scheme in this case is based on downloading MDS-coded mixtures of all messages. For $P \leq \frac{M}{2}$, we develop lower and upper bounds for all $M,P,N$. These bounds match if the total number of messages $M$ is an integer multiple of the number of desired messages $P$, i.e., $\frac{M}{P} \in \mathbb{N}$. In this case, $C_s^P=\frac{1-\frac{1}{N}}{1-(\frac{1}{N})^{M/P}}$. The achievable scheme in this case generalizes the single-message capacity achieving scheme to have unbalanced number of stages per round of download. For all the remaining cases, the difference between the lower and upper bound is at most $0.0082$, which occurs for $M=5$, $P=2$, $N=2$. Our results indicate that joint retrieval of desired messages is more efficient than successive use of single-message retrieval schemes.

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arXiv2016

The Capacity of Private Information Retrieval from Coded Databases

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 $C=\left(1+\frac{K}{N}+\frac{K^2}{N^2}+\cdots+\frac{K^{M-1}}{N^{M-1}}\right)^{-1}=(1+R_c+R_c^2+\cdots+R_c^{M-1})^{-1}=\frac{1-R_c}{1-R_c^M}$, where $R_c$ is the rate of the $(N,K)$ code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases.

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arXiv2016

MIMO Wiretap Channel under Receiver Side Power Constraints with Applications to Wireless Power Transfer and Cognitive Radio

We consider the multiple-input multiple-output (MIMO) wiretap channel under a minimum receiver-side power constraint in addition to the usual maximum transmitter-side power constraint. This problem is motivated by energy harvesting communications with wireless energy transfer, where an added goal is to deliver a minimum amount of energy to a receiver in addition to delivering secure data to another receiver. In this paper, we characterize the exact secrecy capacity of the MIMO wiretap channel under transmitter and receiver-side power constraints. We first show that solving this problem is equivalent to solving the secrecy capacity of the wiretap channel under a double-sided correlation matrix constraint on the channel input. We show the converse by extending the channel enhancement technique to our case. We present two achievable schemes that achieve the secrecy capacity: the first achievable scheme uses a Gaussian codebook with a fixed mean, and the second achievable scheme uses artificial noise (or cooperative jamming) together with a Gaussian codebook. The role of the mean or the artificial noise is to enable energy transfer without sacrificing from the secure rate. This is the first instance of a channel model where either the use of a mean signal or the use of channel prefixing via artificial noise is strictly necessary for the MIMO wiretap channel. We then extend our work to consider a maximum receiver-side power constraint. This problem is motivated by cognitive radio applications, where an added goal is to decrease the received signal energy (interference temperature) at a receiver. We further extend our results to: requiring receiver-side power constraints at both receivers; considering secrecy constraints at both receivers to study broadcast channels with confidential messages; and removing the secrecy constraints to study the classical broadcast channel.

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