### 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 |
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Title of host publication | Proceedings of the Annual ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 504-509 |

Number of pages | 6 |

ISBN (Print) | 0897912640, 9780897912648 |

DOIs | |

Publication status | Published - 1988 |

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

### Other

Other | 20th Annual ACM Symposium on Theory of Computing, STOC 1988 |
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Country | United States |

City | Chicago, IL |

Period | 5/2/88 → 5/4/88 |

### Fingerprint

### ASJC Scopus subject areas

- Software

### Cite this

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

**Two infinite sets of primes with fast primality tests.** / Pintz, Janos; Steiger, William L.; Szemerédi, E.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Annual ACM Symposium on Theory of Computing.*Association for Computing Machinery, pp. 504-509, 20th Annual ACM Symposium on Theory of Computing, STOC 1988, Chicago, IL, United States, 5/2/88. https://doi.org/10.1145/62212.62261

}

TY - GEN

T1 - Two infinite sets of primes with fast primality tests

AU - Pintz, Janos

AU - Steiger, William L.

AU - Szemerédi, E.

PY - 1988

Y1 - 1988

N2 - 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 cn1/3logn 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.

AB - 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 cn1/3logn 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.

UR - http://www.scopus.com/inward/record.url?scp=84898994539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84898994539&partnerID=8YFLogxK

U2 - 10.1145/62212.62261

DO - 10.1145/62212.62261

M3 - Conference contribution

SN - 0897912640

SN - 9780897912648

SP - 504

EP - 509

BT - Proceedings of the Annual ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

ER -