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ACM Symposium on Theory of Computing (STOC)

One-way functions are essential for single-server private information retrieval

Amos Beimel, Yuval Ishai, Eyal Kushilevitz Z, Tal Malkin

Published May 1, 1999• 61 citations
Publisher

Abstract

Private Information Retrieval (PIR) protocols allow a user to read information from a database without revealing to the server storing the database which information he has read.Kushilevitz and Ostrovsky [23] construct, based on the quadratic residuosity assumption, a single-server PIR protc-co1 with small communication complexity.Cachin, Micali, and Stadler [6] present a single-server PIR protocol with a smaller communication complexity, based an the (new) *hiding assumption.A major question, addressed in the present work, is what assumption is the minimal assumption necessary for the construction of single-server private information retrieval protocols with small communication complexity.We prove that if there is a (O-error) PIR protocol in which the server sends less than n bits then one-way functions exist (where n is the number of bits in the database).That is, even saving one bit compared to the naive protocol, in which the entire database is sent, already requires one-way functions.The same result holds (but requires more work) even if we allow the retrieval to fail with probability of at most 1/(8n).Moreover, similar tcomputer science

DOI
10.1145/301250.301277

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We study the communication complexity of single-server Private Information Retrieval (PIR) protocolsthat are based on fundamental cryptographic primitives in a black-box manner. In this setting, we establish a tight lower bound on the number of bits communicated by the server in any polynomially-preservingconstruction that relies on trapdoor permutations. More specifically, our main result states that in such constructions \\Omega (n) bits must be communicated by the server, where n is the size of the server's database,and this improves the \\Omega (n / log n) lower bound due to Haitner, Hoch, Reingold and Segev (FOCS '07).Therefore, in the setting under consideration, the naive solution in which the user downloads the entire database turns out to be optimal up to constant multiplicative factors. We note that the lower boundwe establish holds for the most generic form of trapdoor permutations, including in particular enhanced trapdoor permutations.Technically speaking, this paper consists of two main contributions from which our lower bound is obtained. First, we derive a tight lower bound on the number of bits communicated by the sender duringthe commit stage of any black-box construction of a statistically-hiding bit-commitment scheme from a family of trapdoor permutations. This lower bound asymptotically matches the upper bound provided bythe scheme of Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). Second, we improve the e fficiencyof the reduction of statistically-hiding commitment schemes to low-communication single-server PIR, due to Beimel, Ishai, Kushilevitz and Malkin (STOC '99). In particular, we present a reduction that essentiallypreserves the communication complexity of the underlying single-server PIR protocol.

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A Private Information Retrieval (PIR) protocol allows a client to learn the ith row of a database held by one or more servers, without revealing i to the servers. A Random-Index PIR (RPIR) protocol, introduced by Gentry et al. (TCC 2021), is a PIR protocol where, instead of being chosen by the client, i is random. This has applications in e.g. anonymous committee selection. Both PIR and RPIR protocols are interesting only if the communication complexity is smaller than the database size; otherwise, the trivial solution where the servers send the entire database suffices. Unlike PIR, where the client must send at least one message (to encode information about i), RPIR can be executed in a single round of server-to-client communication. In this paper, we study such one-round, information-theoretic RPIR protocols. The only known construction in this setting is SimpleMSRPIR (Gentry et al.), which requires the servers to communicate approximately N/2 bits, N being the database size. We show an Ω(√N) lower bound on communication complexity for one-round two-server information-theoretic RPIR, and a sublinear upper bound. Finally, we show how to use a sublinear amount of database-independent correlated randomness among multiple servers to get near-optimal online communication complexity (the size of one row plus the size of one index description per server).

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Information-Theoretic Random-Index PIR

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A Private Information Retrieval (PIR) protocol allows a client to learn the ith row of a database held by one or more servers, without revealing i to the servers. A Random-Index PIR (RPIR) protocol, introduced by Gentry et al. (TCC 2021), is a PIR protocol where, instead of being chosen by the client, i is random. This has applications in e.g. anonymous committee selection. Both PIR and RPIR protocols are interesting only if the communication complexity is smaller than the database size; otherwise, the trivial solution where the servers send the entire database suffices. Unlike PIR, where the client must send at least one message (to encode information about i), RPIR can be executed in a single round of server-to-client communication. In this paper, we study such one-round, information-theoretic RPIR protocols. The only known construction in this setting is SimpleMSRPIR (Gentry et al.), which requires the servers to communicate approximately N/2 bits, N being the database size. We show an Ω(√N) lower bound on communication complexity for one-round two-server information-theoretic RPIR, and a sublinear upper bound. Finally, we show how to use a sublinear amount of database-independent correlated randomness among multiple servers to get near-optimal online communication complexity (the size of one row plus the size of one index description per server).

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Private information retrieval (PIR) allows a client to retrieve any block xifrom a database x = x1xnsuch that i remains hidden from the database servers. PIR •protocols with unconditional privacy and sublinear (in n) communication complexity can be constructed assuming multiple honest-but-curious servers. This assumption however cannot be guaranteed in many real life scenarios such as using cloud servers as database servers. In this paper, we consider an information-theoretic PIR with result verification (PIR-RV) model where the servers may be dishonest (i.e., cheating) and provide incorrect answers but the client can detect the existence of cheating servers. We construct a 2-server PIR-RV protocol with communication complexity ${\mathcal{O}}({n^{1/2}}{\text{log }}p)$ where p is a parameter and controls the probability that the client fails to detect. Our idea may be extended to construct k-server PIR-RV protocols for k ≥ 3.

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Efficient transformation capabilities of single database private block retrieval

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Abstract Private information retrieval (PIR) is one of the promising techniques to preserve user privacy in the presence of trusted-but-curious servers. The information theoretically private query construction assures the highest user privacy over curious and unbounded computation servers. Therefore, the need for information-theoretic private retrieval was fulfilled by various schemes in a variety of PIR settings. But, there is a lack of efficient encryption switching scheme which supports efficient switching between information-theoretic to/from computationally bounded PIRs. We propose a combination of new bit connection methods called rail-shape and signal-shape and new quadratic residuosity assumption based family of trapdoor functions for generic single database private block retrieval (PBR). The main goal of this work is to show that the possibility of mapping from computationally bounded privacy to information theoretic privacy or vice-versa in a single database setting using newly constructed bit connection and trapdoor function combinations. Notably, the proposed schemes are single round, memoryless and plain database schemes (at their basic constructions).

International Journal of Information Technology•Jan 19, 2022•4 citations
Private Information Retrieval (PIR) protocols allow a user to read information from a database without revealing to the server storing the database which information he has read.Kushilevitz and Ostrovsky [23] construct, based on the quadratic residuosity assumption, a single-server PIR protc-co1 with small communication complexity.Cachin, Micali, and Stadler [6] present a single-server PIR protocol with a smaller communication complexity, based an the (new) *hiding assumption.A major question, addressed in the present work, is what assumption is the minimal assumption necessary for the construction of single-server private information retrieval protocols with small communication complexity.We prove that if there is a (O-error) PIR protocol in which the server sends less than n bits then one-way functions exist (where n is the number of bits in the database).That is, even saving one bit compared to the naive protocol, in which the entire database is sent, already requires one-way functions.The same result holds (but requires more work) even if we allow the retrieval to fail with probability of at most 1/(8n).Moreover, similar tcomputer science
We study the communication complexity of single-server Private Information Retrieval (PIR) protocolsthat are based on fundamental cryptographic primitives in a black-box manner. In this setting, we establish a tight lower bound on the number of bits communicated by the server in any polynomially-preservingconstruction that relies on trapdoor permutations. More specifically, our main result states that in such constructions \\Omega (n) bits must be communicated by the server, where n is the size of the server's database,and this improves the \\Omega (n / log n) lower bound due to Haitner, Hoch, Reingold and Segev (FOCS '07).Therefore, in the setting under consideration, the naive solution in which the user downloads the entire database turns out to be optimal up to constant multiplicative factors. We note that the lower boundwe establish holds for the most generic form of trapdoor permutations, including in particular enhanced trapdoor permutations.Technically speaking, this paper consists of two main contributions from which our lower bound is obtained. First, we derive a tight lower bound on the number of bits communicated by the sender duringthe commit stage of any black-box construction of a statistically-hiding bit-commitment scheme from a family of trapdoor permutations. This lower bound asymptotically matches the upper bound provided bythe scheme of Naor, Ostrovsky, Venkatesan and Yung (CRYPTO '92). Second, we improve the e fficiencyof the reduction of statistically-hiding commitment schemes to low-communication single-server PIR, due to Beimel, Ishai, Kushilevitz and Malkin (STOC '99). In particular, we present a reduction that essentiallypreserves the communication complexity of the underlying single-server PIR protocol.
A Private Information Retrieval (PIR) protocol allows a client to learn the ith row of a database held by one or more servers, without revealing i to the servers. A Random-Index PIR (RPIR) protocol, introduced by Gentry et al. (TCC 2021), is a PIR protocol where, instead of being chosen by the client, i is random. This has applications in e.g. anonymous committee selection. Both PIR and RPIR protocols are interesting only if the communication complexity is smaller than the database size; otherwise, the trivial solution where the servers send the entire database suffices. Unlike PIR, where the client must send at least one message (to encode information about i), RPIR can be executed in a single round of server-to-client communication. In this paper, we study such one-round, information-theoretic RPIR protocols. The only known construction in this setting is SimpleMSRPIR (Gentry et al.), which requires the servers to communicate approximately N/2 bits, N being the database size. We show an Ω(√N) lower bound on communication complexity for one-round two-server information-theoretic RPIR, and a sublinear upper bound. Finally, we show how to use a sublinear amount of database-independent correlated randomness among multiple servers to get near-optimal online communication complexity (the size of one row plus the size of one index description per server).
A Private Information Retrieval (PIR) protocol allows a client to learn the ith row of a database held by one or more servers, without revealing i to the servers. A Random-Index PIR (RPIR) protocol, introduced by Gentry et al. (TCC 2021), is a PIR protocol where, instead of being chosen by the client, i is random. This has applications in e.g. anonymous committee selection. Both PIR and RPIR protocols are interesting only if the communication complexity is smaller than the database size; otherwise, the trivial solution where the servers send the entire database suffices. Unlike PIR, where the client must send at least one message (to encode information about i), RPIR can be executed in a single round of server-to-client communication. In this paper, we study such one-round, information-theoretic RPIR protocols. The only known construction in this setting is SimpleMSRPIR (Gentry et al.), which requires the servers to communicate approximately N/2 bits, N being the database size. We show an Ω(√N) lower bound on communication complexity for one-round two-server information-theoretic RPIR, and a sublinear upper bound. Finally, we show how to use a sublinear amount of database-independent correlated randomness among multiple servers to get near-optimal online communication complexity (the size of one row plus the size of one index description per server).
Private information retrieval (PIR) allows a client to retrieve any block xifrom a database x = x1xnsuch that i remains hidden from the database servers. PIR •protocols with unconditional privacy and sublinear (in n) communication complexity can be constructed assuming multiple honest-but-curious servers. This assumption however cannot be guaranteed in many real life scenarios such as using cloud servers as database servers. In this paper, we consider an information-theoretic PIR with result verification (PIR-RV) model where the servers may be dishonest (i.e., cheating) and provide incorrect answers but the client can detect the existence of cheating servers. We construct a 2-server PIR-RV protocol with communication complexity O(n1/2log p){\mathcal{O}}({n^{1/2}}{\text{log }}p)O(n1/2log p) where p is a parameter and controls the probability that the client fails to detect. Our idea may be extended to construct k-server PIR-RV protocols for k ≥ 3.
Abstract. In this paper we present a single-round, single-server symmetrically private information retrieval scheme, in which privacy of user follows from intractability of the quadratic residuacity problem and the privacy of the database follows from the XOR assumption for quadratic residues introduced in this paper. The communication complexity of the proposed scheme for retrieving one bit can be made O(n ɛ), for any ɛ>0, where n is the number of bits in the database. We extend the protocol to a block retrieval scheme which is specially efficient when the number of records in the database is equal to the size of each record.
Abstract Private information retrieval (PIR) is one of the promising techniques to preserve user privacy in the presence of trusted-but-curious servers. The information theoretically private query construction assures the highest user privacy over curious and unbounded computation servers. Therefore, the need for information-theoretic private retrieval was fulfilled by various schemes in a variety of PIR settings. But, there is a lack of efficient encryption switching scheme which supports efficient switching between information-theoretic to/from computationally bounded PIRs. We propose a combination of new bit connection methods called rail-shape and signal-shape and new quadratic residuosity assumption based family of trapdoor functions for generic single database private block retrieval (PBR). The main goal of this work is to show that the possibility of mapping from computationally bounded privacy to information theoretic privacy or vice-versa in a single database setting using newly constructed bit connection and trapdoor function combinations. Notably, the proposed schemes are single round, memoryless and plain database schemes (at their basic constructions).