### Abstract

Infinite sets P and Q of primes are described, p ⊂ Q. For any natural number n it can be decided if n∈ p in (deterministic) time O((log n ^{9}). This answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time, and is the main result of the paper. Also, for every n∈Q.we show how to produce at random, in expected time O((log n)^{3}), a certificate of length O(logn) which can be verified in (deterministic) time O((log n) ^{3}); this is less than the time needed for two exponentiations and is much faster than existing methods. Finally it is important that P is relatively dense (at least cn^{1/3}logn elements less than n). Elements of Q in a given range may be generated quickly, but it would be costly for an adversary to search Qin this range; this could be useful in cryptography.

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

Title of host publication | Proceedings of the 20th Annual ACM Symposium on Theory of Computing, STOC 1988 |

Publisher | Association for Computing Machinery |

Pages | 504-509 |

Number of pages | 6 |

ISBN (Print) | 0897912640, 9780897912648 |

DOIs | |

Publication status | Published - Jan 1 1988 |

Event | 20th Annual ACM Symposium on Theory of Computing, STOC 1988 - Chicago, IL, United States Duration: May 2 1988 → May 4 1988 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
---|---|

ISSN (Print) | 0737-8017 |

### Other

Other | 20th Annual ACM Symposium on Theory of Computing, STOC 1988 |
---|---|

Country | United States |

City | Chicago, IL |

Period | 5/2/88 → 5/4/88 |

### ASJC Scopus subject areas

- Software

## Fingerprint Dive into the research topics of 'Two infinite sets of primes with fast primality tests'. Together they form a unique fingerprint.

## Cite this

*Proceedings of the 20th Annual ACM Symposium on Theory of Computing, STOC 1988*(pp. 504-509). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/62212.62261