## 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_{k0} ⊕ F_{k1} = 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(√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).

Original language | English |
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Title of host publication | Advances in Cryptology, EUROCRYPT 2014 - 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings |

Publisher | Springer Verlag |

Pages | 640-658 |

Number of pages | 19 |

ISBN (Print) | 9783642552199 |

DOIs | |

State | Published - 1 Jan 2014 |

Event | 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2014 - Copenhagen, Denmark Duration: 11 May 2014 → 15 May 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8441 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 33rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2014 |
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Country/Territory | Denmark |

City | Copenhagen |

Period | 11/05/14 → 15/05/14 |

## Keywords

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