### Abstract

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a number n, the algorithm chooses "candidates"x_{1}, x_{2}, ..., x_{k} uniformly and independently at random from ℤ_{n}, and tests if any is a "witness" to the compositeness of n. For either algorithm, the probabilty that it errs is at most 2^{-k}. In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen as x, x+1, ..., x+k-1, where x is chosen uniformly at random from ℤ_{n}. We prove that for k=[1/2log_{2}n], the error probability of the Miller-Rabin test is no more than n^{-1/2+o(1)}, which improves on the bound n^{-1/4+o(1)} previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound of n^{-1/2+o(1)} if the number of distinct prime factors of n is o(log n/loglog n).

Original language | English (US) |
---|---|

Pages (from-to) | 355-367 |

Number of pages | 13 |

Journal | Computational Complexity |

Volume | 3 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1993 |

### Fingerprint

### Keywords

- derandomization
- primality
- randomized algorithms
- Subject classifications: 11Y11, 11Y16

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Mathematics
- Mathematics(all)
- Computational Theory and Mathematics

### Cite this

*Computational Complexity*,

*3*(4), 355-367. https://doi.org/10.1007/BF01275488

**Primality testing with fewer random bits.** / Peralta, René; Shoup, Victor.

Research output: Contribution to journal › Article

*Computational Complexity*, vol. 3, no. 4, pp. 355-367. https://doi.org/10.1007/BF01275488

}

TY - JOUR

T1 - Primality testing with fewer random bits

AU - Peralta, René

AU - Shoup, Victor

PY - 1993/12

Y1 - 1993/12

N2 - In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a number n, the algorithm chooses "candidates"x1, x2, ..., xk uniformly and independently at random from ℤn, and tests if any is a "witness" to the compositeness of n. For either algorithm, the probabilty that it errs is at most 2-k. In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen as x, x+1, ..., x+k-1, where x is chosen uniformly at random from ℤn. We prove that for k=[1/2log2n], the error probability of the Miller-Rabin test is no more than n-1/2+o(1), which improves on the bound n-1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound of n-1/2+o(1) if the number of distinct prime factors of n is o(log n/loglog n).

AB - In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a number n, the algorithm chooses "candidates"x1, x2, ..., xk uniformly and independently at random from ℤn, and tests if any is a "witness" to the compositeness of n. For either algorithm, the probabilty that it errs is at most 2-k. In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen as x, x+1, ..., x+k-1, where x is chosen uniformly at random from ℤn. We prove that for k=[1/2log2n], the error probability of the Miller-Rabin test is no more than n-1/2+o(1), which improves on the bound n-1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound of n-1/2+o(1) if the number of distinct prime factors of n is o(log n/loglog n).

KW - derandomization

KW - primality

KW - randomized algorithms

KW - Subject classifications: 11Y11, 11Y16

UR - http://www.scopus.com/inward/record.url?scp=33747885627&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747885627&partnerID=8YFLogxK

U2 - 10.1007/BF01275488

DO - 10.1007/BF01275488

M3 - Article

AN - SCOPUS:33747885627

VL - 3

SP - 355

EP - 367

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 4

ER -