## Abstract

We consider the secure search problem of retrieving from an unsorted data array = (x1, . . ., x_{m} ) an item (i, x_{i} ) matching a given lookup value l (for a generic matching criterion either hardcoded or given as part of the query), where both input and output are encrypted by a Fully Homomorphic Encryption (FHE). The secure search problem is central in applications of secure outsourcing to an untrusted party (“the cloud”). Prior secure search algorithms on FHE encrypted data are realized by polynomials of degree ?(m), evaluated in ?(log m) sequential homomorphic multiplication steps (i.e., multiplicative depth) even with unboundedly many parallel processors. This is too slow with current FHE implementations even for moderate array sizes m such as a few thousands. We present the first secure search algorithm that is realized by a polynomial of logarithmic degree, evaluated in O(log log m) sequential homomorphic multiplication steps (i.e., multiplicative depth) using m parallel processors. We implemented our algorithm in an open source library based on HElib and ran experiments on Amazon’s EC2 cloud with up to 100 processors. Our experiments show that we can securely search in m = millions of entries in less than an hour on a standard EC2 64-cores machine. We achieve our result by: (1) Employing modern data summa-rization techniques known as sketching for returning as output (the encryption of) a short sketch C from which the matching item (i, x_{i} ) can be decoded in time proportional to computing poly(log m) decryption/encryption operations. (2) Designing for this purpose a novel sketch that returns the first strictly-positive entry in a (not necessarily sparse) array of non-negative real numbers; this sketch may be of independent interest. (3) Suggesting a multi-ring evaluation of FHE for degree reduction from linear to logarithmic.

Original language | English |
---|---|

Title of host publication | CCS 2018 - Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security |

Publisher | Association for Computing Machinery |

Pages | 985-1001 |

Number of pages | 17 |

ISBN (Electronic) | 9781450356930 |

DOIs | |

State | Published - 15 Oct 2018 |

Event | 25th ACM Conference on Computer and Communications Security, CCS 2018 - Toronto, Canada Duration: 15 Oct 2018 → … |

### Publication series

Name | Proceedings of the ACM Conference on Computer and Communications Security |
---|---|

ISSN (Print) | 1543-7221 |

### Conference

Conference | 25th ACM Conference on Computer and Communications Security, CCS 2018 |
---|---|

Country/Territory | Canada |

City | Toronto |

Period | 15/10/18 → … |

### Bibliographical note

Funding Information:We thank Shai Halevi, Craig Gentry, Shafi Goldwasser and Vinod Vaikuntanathan as well as the anonymous referees for helpful discussions and comments. The second author is grateful for the support of the Simons Foundation for part of this work that was done while he was visiting the Simons Institute for the Theory of Computing. Part of this work was sponsored by the Center for Cyber Law & Policy in the University of Haifa.

Publisher Copyright:

© 2018 Association for Computing Machinery.

## Keywords

- First positive sketch
- Fully homomorphic encryption
- Homomorphic encryption
- Low degree polynomials
- Secure search
- Sketching

## ASJC Scopus subject areas

- Software
- Computer Networks and Communications