## Priebel/176bit

Quadratic sieve; factoring a 176bit number. D. Priebel, Tenn. Tech Univ

Name | 176bit |
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Group | Priebel |

Matrix ID | 2253 |

Num Rows | 7,441 |

Num Cols | 7,431 |

Nonzeros | 82,270 |

Pattern Entries | 82,270 |

Kind | Combinatorial Problem |

Symmetric | No |

Date | 2009 |

Author | D. Priebel |

Editor | T. Davis |

SVD Statistics | |
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Matrix Norm | 9.619242e+01 |

Minimum Singular Value | 1.319478e-47 |

Condition Number | 7.290185e+48 |

Rank | 7,110 |

sprank(A)-rank(A) | 0 |

Null Space Dimension | 321 |

Full Numerical Rank? | no |

Download Singular Values | MATLAB |

Download | MATLAB Rutherford Boeing Matrix Market |
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Notes |
Each column in the matrix corresponds to a number in the factor base less than some bound B. Each row corresponds to a smooth number (able to be completely factored over the factor base). Each value in a row binary vector corresponds to the exponent of the factor base mod 2. For example: factor base: 2 7 23 smooth numbers: 46, 28, 322 2^1 * 23^1 = 46 2^2 * 7^1 = 28 2^1 * 7^1 * 23^1 = 322 Matrix: 101 010 111 A solution to the matrix is considered to be a set of rows which when combined in GF2 produce a null vector. Thus, if you multiply each of the smooth numbers which correspond to that particular set of rows you will get a number with only even exponents, making it a perfect square. In the above example you can see that combining the 3 vectors results in a null vector and, indeed, it is a perfect square: 644^2. Problem.A: A GF(2) matrix constructed from the exponents of the factorization of the smooth numbers over the factor base. A solution of this matrix is a kernel (nullspace). Such a solution has a 1/2 chance of being a factorization of N. Problem.aux.factor_base: The factor base used. factor_base(j) corresponds to column j of the matrix. Note that a given column may or may not have nonzero elements in the matrix. Problem.aux.smooth_number: The smooth numbers, smooth over the factor base. smooth_number(i) corresponds to row i of the matrix. Problem.aux.solution: A sample solution to the matrix. Combine, in GF(2) the rows with these indicies to produce a solution to the matrix with the additional property that it factors N (a matrix solution only has 1/2 probability of factoring N). Problem specific information: n = 73363722971930954428433124842779099222294372095286387 (176-bits) passes primality test, n is composite, continuing... 1) Initial bound: 180000, pi(180000) estimate: 14875, largest found: 162359 (actual bound) 2) Number of quadratic residues estimate: 9918, actual number found: 7431 3) Modular square roots found: 14862(2x residues) 4) Constructing smooth number list [sieving] (can take a while)... Sieving for: 7441 5. Constructing a matrix of size: 7441x7432 Set a total of 82270 exponents, with 3725 negatives Matrix solution found with: 2983 combinations Divisor: 236037985789994529800050193 (probably prime) Divisor: 310813205452462332837525059 (probably prime) |