## Abstract

This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat-Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space S and (online) time T for a preprocessing version of the N-input 3SUM problem where S^{3}· T = O(N^{6}). This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for S=N^{2-} and T = N^{1-} for any constant >0. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function F: [N] → [N] (accessed as an oracle) we show how to compile it into a function G^{F}: [N^{2}] → [N^{2}] which resists S-bit preprocessing attacks that run in query time T where ST=O(N^{2-ϵ}) (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that F itself can be more easily inverted, say with N^{2/3}-bit preprocessing in N^{2/3} time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM which match the best known lower bounds for static data structure problems. Moreover, we show that our lower bound generalizes to a range of geometric problems, such as three points on a line, polygon containment, and others.

Original language | English (US) |
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Title of host publication | STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy |

Publisher | Association for Computing Machinery |

Pages | 294-307 |

Number of pages | 14 |

ISBN (Electronic) | 9781450369794 |

DOIs | |

State | Published - Jun 8 2020 |

Event | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, United States Duration: Jun 22 2020 → Jun 26 2020 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 |
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Country/Territory | United States |

City | Chicago |

Period | 6/22/20 → 6/26/20 |

## Keywords

- Cryptography with preprocessing
- Data structures
- Fine-grained complexity

## ASJC Scopus subject areas

- Software