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Register a new userXOR-Based Codes for Private Information Retrieval with Private Side Information
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Added by
Jagadeesh Harshan
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We consider the problem of Private Information Retrieval with Private Side Information (PIR-PSI), wherein a user wants to retrieve a file from replication based non-colluding databases by using the prior knowledge of a subset of the files stored on the databases. The PIR-PSI framework ensures that the privacy of the demand and the side information are jointly preserved, thereby finding potential applications when multiple files have to be downloaded spread across different time-instants. Although the capacity of the PIR-PSI setting is known, we observe that the underlying capacity achieving code construction uses Maximum Distance Separable (MDS) codes thereby contributing to high computational complexity when retrieving the demand. Pointing at this drawback of MDS-based PIR-PSI codes, we propose XOR-based PIR-PSI codes for a simple yet non-trivial setting of two non-colluding databases and two side information files at the user. While our codes offer substantial reduction in complexity when compared to MDS based codes, the code-rate marginally falls short of the capacity of the PIR-PSI setting. Nevertheless, we show that our code-rate is strictly higher than that of XOR-based codes for PIR with no side information, thereby implying that our codes can be useful when downloading multiple files in a sequential manner, instead of applying XOR-based PIR codes on each file.
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In the conventional robust $T$-colluding private information retrieval (PIR) system, the user needs to retrieve one of the possible messages while keeping the identity of the requested message private from any $T$ colluding servers. Motivated by the possible heterogeneous privacy requirements for different messages, we consider the $(N, T_1:K_1, T_2:K_2)$ two-level PIR system, where $K_1$ messages need to be retrieved privately against $T_1$ colluding servers, and all the messages need to be retrieved privately against $T_2$ colluding servers where $T_2leq T_1$. We obtain a lower bound to the capacity by proposing two novel coding schemes, namely the non-uniform successive cancellation scheme and the non-uniform block cancellation scheme. A capacity upper bound is also derived. The gap between the upper bound and the lower bounds is analyzed, and shown to vanish when $T_1=T_2$. Lastly, we show that the upper bound is in general not tight by providing a stronger bound for a special setting.
The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with replicated-server PIR. In the present paper, the construction of an $(n,k)$ $tau$-server binary, linear PIR code having parameters $n = sumlimits_{i = 0}^{ell} {m choose i}$, $k = {m choose ell}$ and $tau = 2^{ell}$ is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary, Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any $tau$ of the form $2^{ell}$, $2^{ell}-1$ and any value of $k$, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with other PIR codes appearing in the literature. For the particular case of $tau=3,4$, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length $n(k, tau)$ of a systematic PIR code. It follows from a result by Vardy and Yaakobi, that these codes also yield optimal, systematic primitive multi-set $(n, k, tau)_B$ batch codes for $tau=3,4$. The PIR code constructions presented here also yield upper bounds on the generalized Hamming weights of binary PRM codes.
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 and quotient error search problem, which we assume to be difficult, and show that the scheme is secure against attacks based on solving these problems.
We investigate the problem of semantic private information retrieval (semantic PIR). In semantic PIR, a user retrieves a message out of $K$ independent messages stored in $N$ replicated and non-colluding databases without revealing the identity of the desired message to any individual database. The messages come with emph{different semantics}, i.e., the messages are allowed to have emph{non-uniform a priori probabilities} denoted by $(p_i>0,: i in [K])$, which are a proxy for their respective popularity of retrieval, and emph{arbitrary message sizes} $(L_i,: i in [K])$. This is a generalization of the classical private information retrieval (PIR) problem, where messages are assumed to have equal a priori probabilities and equal message sizes. We derive the semantic PIR capacity for general $K$, $N$. The results show that the semantic PIR capacity depends on the number of databases $N$, the number of messages $K$, the a priori probability distribution of messages $p_i$, and the message sizes $L_i$. We present two achievable semantic PIR schemes: The first one is a deterministic scheme which is based on message asymmetry. This scheme employs non-uniform subpacketization. The second scheme is probabilistic and is based on choosing one query set out of multiple options at random to retrieve the required message without the need for exponential subpacketization. We derive necessary and sufficient conditions for the semantic PIR capacity to exceed the classical PIR capacity with equal priors and sizes. Our results show that the semantic PIR capacity can be larger than the classical PIR capacity when longer messages have higher popularities. However, when messages are equal-length, the non-uniform priors cannot be exploited to improve the retrieval rate over the classical PIR capacity.
We introduce the problem of emph{timely} private information retrieval (PIR) from $N$ non-colluding and replicated servers. In this problem, a user desires to retrieve a message out of $M$ messages from the servers, whose contents are continuously updating. The retrieval process should be executed in a timely manner such that no information is leaked about the identity of the message. To assess the timeliness, we use the emph{age of information} (AoI) metric. Interestingly, the timely PIR problem reduces to an AoI minimization subject to PIR constraints under emph{asymmetric traffic}. We explicitly characterize the optimal tradeoff between the PIR rate and the AoI metric (peak AoI or average AoI) for the case of $N=2$, $M=3$. Further, we provide some structural insights on the general problem with arbitrary $N$, $M$.
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