Sphere Packings, Lattices and Groups

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 $T$ of the $N$ databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have $M \geq N$ databases, out of which any $M - N$ may fail to respond. For $K$ messages and $M\geq N$ databases out of which at least some $N$ must respond, we show that the capacity of $T$ -private and Robust PIR is $(1+T/N+T^{2}/N^{2}+\cdots +T^{K-1}/N^{K-1})^{-1}$ . The result includes as special cases the capacity of PIR without robustness ( $M=N$ ) or $T$ -privacy constraints ( $T=1$ ).

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The Capacity of Robust Private Information Retrieval With Colluding Databases

We say that a network coding scheme is strongly 1-secure if a source node s can multicast n field elements {m1, · · · ,mn} to a set of sink nodes {t1, · · · , tq} in such a way that any single edge leaks no information on any S ⊂ {m1, · · · ,mn} with |S| = n − 1, where n = mintimax-flow(s, ti) is the maximum transmission capacity. We also say that a strongly h-secure network coding scheme is strongly (h + 1)secure if any h + 1 edges leak no information on any S ⊂ {m1, · · · ,mn} with |S| = n − (h + 1). In this paper, we show the first explicit algorithm which can construct strongly k-secure network coding schemes. In particular, it runs in polynomial time for fixed k.

Information Theoretic Security

The problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS), is considered. The DSS is coded by an $(n,k,d)$ maximum distance separable code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing no information on which data is being requested to the nodes. A user can trivially achieve PIR by downloading all the data in the DSS. However, this is not a feasible solution due to its high communication cost. We construct PIR schemes with low download communication cost. When there is $b=1$ spy node in the DSS, in other words, no collusion between the nodes, we construct PIR schemes with download cost $\frac {1}{1-R}$ per unit of requested data ( $R=k/n$ is the code rate), achieving the information theoretic limit for linear schemes. The proposed schemes are universal since they depend on the code rate, but not on the generator matrix of the code. Also, if $b\leq n-\delta k$ nodes collude, with $\delta =\lfloor {\frac {n-b}{k}}\rfloor $ , we construct linear PIR schemes with download cost $\frac {b+\delta k}{\delta }$ .

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Private Information Retrieval From MDS Coded Data in Distributed Storage Systems

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 information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is $1-1/N$ regardless of the number of messages $K$ , if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least $1/(N-1)$ bits per desired message bit. Otherwise, if the amount of common randomness is less than $1/(N-1)$ bits per message bit, then the capacity of SPIR is zero. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided.

The Capacity of Symmetric Private Information Retrieval

We present a general framework for private information retrieval (PIR) from arbitrary coded databases that allows one to adjust the rate of the scheme to the suspected number of colluding servers. ...

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Private Information Retrieval from Coded Databases with Colluding Servers

The problem of providing privacy, in the private information retrieval (PIR) sense, to users requesting data from a distributed storage system (DSS), is considered. The DSS is coded by an $(n,k,d)$ Maximum Distance Separable (MDS) code to store the data reliably on unreliable storage nodes. Some of these nodes can be spies which report to a third party, such as an oppressive regime, which data is being requested by the user. An information theoretic PIR scheme ensures that a user can satisfy its request while revealing, to the spy nodes, no information on which data is being requested. A user can trivially achieve PIR by downloading all the data in the DSS. However, this is not a feasible solution due to its high communication cost. We construct PIR schemes with low download communication cost. When there is $b=1$ spy node in the DSS, we construct PIR schemes with download cost $\frac{1}{1-R}$ per unit of requested data ($R=k/n$ is the code rate), achieving the information theoretic limit for linear schemes. The proposed schemes are universal since they depend on the code rate, but not on the generator matrix of the code. Also, when $b\leq n-\delta k$, for some $\delta \in \mathbb{N^+}$, we construct linear PIR schemes with $cPoP = \frac{b+\delta k}{\delta}$.

Private Information Retrieval from MDS Coded Data in Distributed Storage Systems

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size t colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.

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Private information retrieval schemes for codec data with arbitrary collusion patterns

The problem of private information retrieval (PIR) from coded storage systems with colluding, Byzantine, and unresponsive servers is considered. An explicit scheme using an $[n,k]$ Reed–Solomon storage code is designed, protecting against $t$ -collusion, and handling up to $b$ Byzantine and $r$ unresponsive servers, when $n>k+t+2b+r-1$ . This scheme achieves a PIR rate of $(({n-r-(k+2b+t-1)})/{n-r})$ . In the case where the capacity is known, namely, when $k=1$ , it is asymptotically capacity achieving as the number of files grows. Finally, the scheme is adapted to symmetric PIR.

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Oliver W. Gnilke

Papers

Private Information Retrieval From MDS Coded Data in Distributed Storage Systems

Private Information Retrieval from Coded Databases with Colluding Servers

Private Information Retrieval from MDS Coded Data in Distributed Storage Systems

Private information retrieval schemes for codec data with arbitrary collusion patterns

Private Information Retrieval From Coded Storage Systems With Colluding, Byzantine, and Unresponsive Servers