## Abstract

Motivated by the goal of securely searching and updating distributed data, we introduce and study the notion of function secret sharing (FSS). This new notion is a natural generalization of distributed point functions (DPF), a primitive that was recently introduced by Gilboa and Ishai (Eurocrypt 2014). Given a positive integer p ≥ 2 and a class F of functions f: {0, 1}^{n} → (image found), where (image found) is an Abelian group, a p-party FSS scheme for F allows one to split each f ∈ F into p succinctly described functions f_{i}: {0, 1}^{n} →(image found), 1 ≤ i ≤ p, such that: (1) ∑^{p} _{i=1} f_{i} = f, and (2) any strict subset of the f_{i} hides f. Thus, an FSS for F can be thought of as method for succinctly performing an “additive secret sharing” of functions from F. The original definition of DPF coincides with a twoparty FSS for the class of point functions, namely the class of functions that have a nonzero output on at most one input. We present two types of results. First, we obtain efficiency improvements and extensions of the original DPF construction. Then, we initiate a systematic study of general FSS, providing some constructions and establishing relations with other cryptographic primitives. More concretely, we obtain the following main results: – Improved DPF. We present an improved (two-party) DPF construction from a pseudorandom generator (PRG), reducing the length of the key describing each fi from O(λ ・ n^{log2 3}) to O(λn), where λ is the PRG seed length. – Multi-party DPF. We present the first nontrivial construction of a p-party DPF for p ≥ 3, obtaining a near-quadratic improvement over a naive construction that additively shares the truth-table of f. This constrcution too can be based on any PRG. – FSS for simple functions. We present efficient PRG-based FSS constructions for natural function classes that extend point functions, including interval functions and partial matching functions. – A study of general FSS. We show several relations between general FSS and other cryptographic primitives. These include a construction of general FSS via obfuscation, an indication for the implausibility of constructing general FSS from weak cryptographic assumptions such as the existence of one-way functions, a completeness result, and a relation with pseudorandom functions.

Original language | English |
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Title of host publication | Advances in Cryptology - 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2015, Proceedings |

Editors | Marc Fischlin, Elisabeth Oswald |

Publisher | Springer Verlag |

Pages | 337-367 |

Number of pages | 31 |

ISBN (Print) | 9783662468029 |

DOIs | |

State | Published - 1 Jan 2015 |

Event | 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2015 - Sofia, Bulgaria Duration: 26 Apr 2015 → 30 Apr 2015 |

### 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 | 9057 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Eurocrypt 2015 |
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Country/Territory | Bulgaria |

City | Sofia |

Period | 26/04/15 → 30/04/15 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)