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Distributed Point Functions and Their Applications

  • Niv Gilboa
  • Yuval Ishai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8441)

Abstract

For x,y ∈ {0,1}*, the point function P x,y is defined by P x,y (x) = y and P x,y (x′) = 0|y| for all x′ ≠ x. We introduce the notion of a distributed point function (DPF), which is a keyed function family F k with the following property. Given x,y specifying a point function, one can efficiently generate a key pair (k 0,k 1) such that: (1) \(F_{k_0}\oplus F_{k_1}=P_{x,y}\), and (2) each of k 0 and k 1 hides x and y. Our main result is an efficient construction of a DPF under the (minimal) assumption that a one-way function exists.

Distributed point functions have applications to private information retrieval (PIR) and related problems, as well as to worst-case to average-case reductions. Concretely, assuming the existence of a strong one-way function, we obtain the following applications.

  • Polylogarithmic 2-server binary PIR. We present the first 2- server computational PIR protocol in which the length of each query is polylogarithmic in the database size n and the answers consist of a single bit each. This improves over the 2\(^O(\sqrt{log n})\) query length of the protocol of Chor and Gilboa (STOC ’97). Similarly, we get a polylogarithmic “PIR writing” scheme, allowing secure non-interactive updates of a database shared between two servers. Assuming just a standard one-way function, we get the first 2-server private keyword search protocol in which the query length is polynomial in the keyword size, the answers consist of a single bit, and there is no error probability. In all these protocols, the computational cost on the server side is comparable to applying a symmetric encryption scheme to the entire database.

  • Worst-case to average-case reductions. We present the first worst-case to average-case reductions for PSPACE and EXPTIME complete languages that require only a constant number of oracle queries. These reductions complement a recent negative result of Watson (TOTC ’12).

Keywords

Distributed point function PIR secure keyword search worst-case to average-case reductions 

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Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Niv Gilboa
    • 1
  • Yuval Ishai
    • 2
  1. 1.Dept. of Communication Systems Eng.Ben-Gurion UniversityBeer-ShevaIsrael
  2. 2.Dept. of Computer ScienceTechnionIsrael

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