## The Quadratic Sieve Factoring Algorithm (2001)

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### BibTeX

@MISC{Landquist01thequadratic,

author = {Eric Landquist},

title = {The Quadratic Sieve Factoring Algorithm},

year = {2001}

}

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### Abstract

Mathematicians have been attempting to find better and faster ways to factor composite numbers since the beginning of time. Initially this involved dividing a number by larger and larger primes until you had the factorization. This trial division was not improved upon until Fermat applied the

### Citations

183 |
Solving sparse linear equations over finite fields
- Wiedemann
- 1986
(Show Context)
Citation Context ... is no possible way for that Q(x) to be a factor in a square. There are two algorithms which do Guassian elimination of a matrix over a finite field: Wiedemann and Lanczos [10]. Of the two, Wiedemann =-=[11]-=- works better over GF (2), which is the field we are in of course. The running time is approximately O(B(w + Bln(B)ln(ln(B))). where B is the number of primes in the factor base, and w is approximatel... |

38 | Factorization and primality testing - Bressoud - 1989 |

21 |
Prime numbers and computer methods for factorization (2nd
- Riesel
- 1994
(Show Context)
Citation Context ...wo Q(xi) will have the factor L 2 . So we add these two factorizations to the matrix A. There may be other Q(xi) which factor over this larger factor base, so we add those in as well. It is estimated =-=[7]-=- that this cuts the sieving time by a sixth. 6 Gaussian Elimination A critical step in the factoring process is the Gaussian elimination step. The matrix that is formed is huge, and almost every entry... |

12 | Theory with Computer Applications - Kumanduri, Romero - 2001 |

5 | Factoring large numbers with a quadratic sieve - Gerver - 1983 |

3 |
et al. Factorization of a 512-bit RSA Modulus. Eurocrypt
- Cavallar
- 2000
(Show Context)
Citation Context ...tored. In August, 1999, a team including Arjen Lenstra and Peter Montgomery factored a 512 bit RSA modulus using the Number Field Sieve in 8400 mips years (8400 million instructions per second-years) =-=[2]-=-. Current estimates say that a 768 bit modulus will be good until 2004, so for short term or personal use, such a key size is adequate. For corporate use, a 1024 bit modulua is suggested, and a 2048 b... |

3 |
Cryptology and Computational Number Theory; Factoring
- Pomerance
- 1990
(Show Context)
Citation Context ...sly our maximum is at −M or M, and is roughly a2M 2−n. a We want this to be about n/a, so we choose √ 2n a ≈ M . One cause for concern with this method is the cost of switching polynomials. Pomerance =-=[6]-=- says that if the cost of switching polynomials is about 25-30% of the total cost, then it would be disadvantageous to use this method. When changing a polynomial, we obviously need new coefficients, ... |

2 | Number Theory for Computing - Song - 2000 |

2 |
Algebra Methods in Cryptography,” unpublished paper
- Webster
- 2001
(Show Context)
Citation Context ...ow associated with it. There is no possible way for that Q(x) to be a factor in a square. There are two algorithms which do Guassian elimination of a matrix over a finite field: Wiedemann and Lanczos =-=[10]-=-. Of the two, Wiedemann [11] works better over GF (2), which is the field we are in of course. The running time is approximately O(B(w + Bln(B)ln(ln(B))). where B is the number of primes in the factor... |

1 | What’s Happening in hte - Cipra - 1996 |

1 |
The Multiple Polynomial Quadratic Sieve Method
- Silverman
- 1987
(Show Context)
Citation Context ...f the primes, subtracting the logarithms as above. Our threshold will then be 1 ln(n) + ln(M) − T ln(pmax) 2 where T is some value around 2 and pmax is the largest prime in the factor base. Silverman =-=[8]-=- suggested that T = 1.5 for factoring 30-digit numbers, T = 2 for 45-digit numbers, and T = 2.6 for 66-digit numbers, for example. 4s3.3 Building the Matrix If Q(x) does completely factor, then we put... |