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1. An Optimal Iterative Placement Algorithm for PIR from Heterogeneous Storage-Constrained Databases

   https://doi.org/10.1109/GLOBECOM38437.2019.9013430  

   Woolsey, Nicholas; Chen, Rong-Rong; Ji, Mingyue (February 2020, 2019 IEEE Global Communications Conference (GLOBECOM))

   We propose a capacity-achieving scheme for private information retrieval (PIR) from databases (DBs) with 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. Our PIR scheme uses an uncoded storage placement and we derive sufficient conditions to meet capacity in this design architecture. We translate the storage placement design to a "filling problem" where messages are partitioned into sub- messages and stored at subsets of DBs. We prove a set of necessary and sufficient conditions for the existence of the filling problem solution and design an iterative algorithm to find a filling problem solution. Our proposed algorithm requires at most a number of iterations equal to the number of DBs. Furthermore, we significantly reduce the number of sub-messages compared to the state-of- the-art PIR scheme, as our proposed PIR scheme requires that each message is split into a polynomial number of sub-messages with respect to the number of DBs. 

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2. The Capacity of Uncoded Storage Constrained PIR

   https://doi.org/10.1109/ISIT.2018.8437729  

   Attia, Mohamed Adel; Kumar, Deepak; Tandon, Ravi (June 2018, 2018 IEEE International Symposium on Information Theory (ISIT))

   Private information retrieval (PIR) allows a user to retrieve a desired message out of K possible messages from N databases (DBs) without revealing the identity of the desired message. In this work, we consider the problem of PIR from uncoded storage constrained DBs. Each DB has a storage capacity of μKL bits, where L is the size of each message in bits, and μ ∈ [1/N, 1] is the normalized storage. In the storage constrained PIR problem, there are two key challenges: a) construction of communication efficient schemes through storage content design at each DB that allow download efficient PIR; and b characterizing the optimal download cost via information-theoretic lower bounds. The novel aspect of this work is to characterize the optimum download cost of PIR with storage constrained DBs for any value of storage. In particular, for any (N, K), we show that the optimal tradeoff between storage (μ) and the download cost (D(μ)) is given by the lower convex hull of the pairs ([t/N](1+[1/t]+[1/(t 2 )]+...+[1/(t K-1 )])) for t = 1,2, ..., N. The main contribution of this paper is the converse proof, i.e., obtaining lower bounds on the download cost for PIR as a function of the available storage. 

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3. PIR from Storage Constrained Databases - Coded Caching Meets PIR

   https://doi.org/10.1109/ICC.2018.8422198  

   Tandon, Ravi; Abdul-Wahid, Maryam; Almoualem, Firas; Kumar, Deepak (May 2018, 2018 IEEE International Conference on Communications (ICC))

   Private information retrieval (PIR) allows a user to retrieve a desired message out of K possible messages from N databases without revealing the identity of the desired message. There has been significant recent progress on understanding fundamental information-theoretic limits of PIR, and in particular the download cost of PIR for several variations. Majority of existing works however, assume the presence of replicated databases, each storing all the K messages. In this work, we consider the problem of PIR from storage constrained databases. Each database has a storage capacity of μKL bits, where K is the number of messages, L is the size of each message in bits, and μ ∈ [1/N, 1] is the normalized storage. In the storage constrained PIR problem, there are two key design questions: a) how to store content across each database under storage constraints; and b) construction of schemes that allow efficient PIR through storage constrained databases. The main contribution of this work is a general achievable scheme for PIR from storage constrained databases for any value of storage. In particular, for any (N, K), with normalized storage μ = t/N, where the parameter t can take integer values t ∈ {1, 2, ..., N}, we show that our proposed PIR scheme achieves a download cost of (1 + 1/t + 1/2 + ⋯ + 1/t K-1 ). The extreme case when μ = 1 (i.e., t = N) corresponds to the setting of replicated databases with full storage. For this extremal setting, our scheme recovers the information-theoretically optimal download cost characterized by Sun and Jafar as (1 + 1/N + ⋯ +1/N K-1 ). For the other extreme, when μ = 1/N (i.e., t = 1), the proposed scheme achieves a download cost of K. The most interesting aspect of the result is that for intermediate values of storage, i.e., 1/N <; μ <; 1, the proposed scheme can strictly outperform memory-sharing between extreme values of storage. 

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4. On Single Server Private Information Retrieval With Private Coded Side Information

   https://doi.org/10.1109/TIT.2023.3253078  

   Lu, Yuxiang; Jafar, Syed A. (May 2023, IEEE Transactions on Information Theory)

   Matthieu Bloch (Ed.)

   Motivated by an open problem and a conjecture, this work studies the problem of single server private information retrieval with private coded side information (PIR-PCSI) that was recently introduced by Heidarzadeh et al. The goal of PIR-PCSI is to allow a user to efficiently retrieve a desired message Wθ, which is one of K independent messages that are stored at a server, while utilizing private side information of a linear combination of a uniformly chosen size-M subset (S ⊂ [K]) of messages. The settings PIR-PCSI-I and PIR-PCSI-II correspond to the constraints that θ is generated uniformly from [K]\S, and S, respectively. In each case, (θ, S) must be kept private from the server. The capacity is defined as the supremum over message and field sizes, of achievable rates (number of bits of desired message retrieved per bit of download) and is characterized by Heidarzadeh et al. for PIR-PCSI-I in general, and for PIR- PCSI-II for M > (K + 1)/2 as (K − M + 1)−1. For 2 ≤ M ≤ (K + 1)/2 the capacity of PIR-PCSI-II remains open, and it is conjectured that even in this case the capacity is (K − M + 1)−1. We show the capacity of PIR-PCSI-II is equal to 2/K for 2 ≤ M ≤ K+1, which is strictly larger 2 than the conjectured value, and does not depend on M within this parameter regime. Remarkably, half the side-information is found to be redundant. We also characterize the infimum capacity (infimum over fields instead of supremum), and the capacity with private coefficients. The results are generalized to PIR-PCSI-I (θ ∈ [K] \ S) and PIR-PCSI (θ ∈ [K]) settings. 

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5. Weakly Private Information Retrieval from Heterogeneously Trusted Servers

   https://doi.org/10.1109/ISIT57864.2024.10619321  

   Huang, Yu-Shin; Zhao, Wenyuan; Zhou, Ruida; Tian, Chao (July 2024, IEEE)

   We study the problem of weakly private information retrieval (PIR) when there is heterogeneity in servers’ trustfulness under the maximal leakage (Max-L) metric. A user wishes to retrieve a desired message from N non-colluding servers efficiently, such that the identity of the desired message is not leaked in a significant manner; however, some servers can be more trustworthy than others. We propose a code construction for this setting and optimize the probability distribution for this construction. It is shown that the optimal probability allocation for the proposed scheme essentially separates the delivery patterns into two parts: a completely private part that has the same download overhead as the capacity-achieving PIR code, and a non-private part that allows complete privacy leakage but has no download overhead by downloading only from the most trustful server. The optimal solution is established through a sophisticated analysis of the underlying convex optimization problem, and a reduction between the homogeneous setting and the heterogeneous setting. 

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