In cryptography, a private information retrieval (PIR) protocol is a protocol that allows a user to retrieve an item from a server in possession of a database without revealing which item is retrieved. PIR is a weaker version of 1-out-of-n oblivious transfer, where it is also required that the user should not get information about other database items. One trivial, but very inefficient way to achieve PIR is for the server to send an entire copy of the database to the user. In fact, this is the only possible protocol (in the classical or the quantum setting) that gives the user information-theoretic privacy for their query in a single-server setting. There are two ways to address this problem: make the server computationally bounded or assume that there are multiple non-cooperating servers, each having a copy of the database. The problem was introduced in 1995 by Chor, Goldreich, Kushilevitz and Sudan in the information-theoretic setting and in 1997 by Kushilevitz and Ostrovsky in the computational setting. Since then, very efficient solutions have been discovered. Single database (computationally private) PIR can be achieved with constant (amortized) communication and k-database (information theoretic) PIR can be done with n O ( log log k k log k ) {\displaystyle n^{O\left({\frac {\log \log k}{k\log k}}\right)}} communication.
Advances in computational PIR The first single-database computational PIR scheme to achieve communication complexity less than n {\displaystyle n} was created in 1997 by Kushilevitz and Ostrovsky and achieved communication complexity of n ϵ {\displaystyle n^{\epsilon }} for any ϵ {\displaystyle \epsilon } , where n {\displaystyle n} is the number of bits in the database. The security of their scheme was based on the well-studied quadratic residuosity problem. In 1999, Christian Cachin, Silvio Micali and Markus Stadler achieved poly-logarithmic communication complexity. The security of their system is based on the phi-hiding assumption. In 2004, Helger Lipmaa achieved log-squared communication complexity O ( ℓ log n + k log 2 n ) {\displaystyle O(\ell \log n+k\log ^{2}n)} , where ℓ {\displaystyle \ell } is the length of the strings and k {\displaystyle k} is the security parameter. The security of his system reduces to the semantic security of a length-flexible additively homomorphic cryptosystem like the Damgård–Jurik cryptosystem. In 2005 Craig Gentry and Zulfikar Ramzan achieved log-squared communication complexity which retrieves log-square (consecutive) bits of the database. The security of their scheme is also based on a variant of the Phi-hiding assumption. The communication rate was finally brought down to 1 {\displaystyle 1} by Aggelos Kiayias, Nikos Leonardos, Helger Lipmaa, Kateryna Pavlyk, Qiang Tang, in 2015. All previous sublinear-communication computational PIR protocol required linear computational complexity of Ω ( n ) {\displaystyle \Omega (n)} public-key operations. In 2009, Helger Lipmaa designed a computational PIR protocol with communication complexity O ( ℓ log n + k log 2 n ) {\displaystyle O(\ell \log n+k\log ^{2}n)} and worst-case computation of O ( n / log n ) {\displaystyle O(n/\log n)} public-key operations. Amortization techniques that retrieve non-consecutive bits have been considered by Yuval Ishai, Eyal Kushilevitz, Rafail Ostrovsky and Amit Sahai. As shown by Ostrovsky and Skeith, the schemes by Kushilevitz and Ostrovsky and Lipmaa use similar ideas based on homomorphic encryption. The Kushilevitz and Ostrovsky protocol is based on the Goldwasser–Micali cryptosystem while the protocol by Lipmaa is based on the Damgård–Jurik cryptosystem.
Advances in information theoretic PIR Achieving information theoretic security requires the assumption that there are multiple non-cooperating servers, each having a copy of the database. Without this assumption, any information-theoretically secure PIR protocol requires an amount of communication that is at least the size of the database n. Multi-server PIR protocols tolerant of non-responsive or malicious/colluding servers are called robust or Byzantine robust respectively. These issues were first considered by Beimel and Stahl (2002). An ℓ-server system that can operate where only k of the servers respond, ν of the servers respond incorrectly, and which can withstand up to t colluding servers without revealing the client's query is called "t-private ν-Byzantine robust k-out-of-ℓ PIR" [DGH 2012]. In 2012, C. Devet, I. Goldberg, and N. Heninger (DGH 2012) proposed an optimally robust scheme that is Byzantine-robust to ν < k − t − 1 {\displaystyle \nu <k-t-1} which is the theoretical maximum value. It is based on an earlier protocol of Goldberg that uses Shamir's Secret Sharing to hide the query. Goldberg has released a C++ implementation on SourceForge.
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