## Group Priebel

Group Description |
Quadratic sieve matrices from David Priebel, Tenn. Tech. Univ. 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 = 787911249838484617926390474950774839909 (130-bits) Divisor: 23850290715477455017 (probably prime) Divisor: 33035708421246870877 (probably prime) n = 27393004579711727757848513391018843988362569 (145-bits) Divisor: 4762476283061573160587 (probably prime) Divisor: 5751840629031342254587 (probably prime) n = 3408489886335277144344023699527218196631767672957 (162-bits) Divisor: 1816046478796474796528999 (probably prime) Divisor: 1876873706775468629074043 (probably prime) n = 73363722971930954428433124842779099222294372095286387 (176-bits) Divisor: 236037985789994529800050193 (probably prime) Divisor: 310813205452462332837525059 (probably prime) n = 4232562527578032866150921497850842593296760823796443077101 (192-bits) Divisor: 83135929635332984850508004533 (probably prime) Divisor: 50911351399373573113182167897 (probably prime) n = 239380926372595066574100671394554319947805305453767699448870971 (208-bits) Divisor: 12216681953629826483019726942851 (probably prime) Divisor: 19594594283554225566102143686121 (probably prime) |
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Displaying

**all 6**collection matricesId | Name | Group | Rows | Cols | Nonzeros | Kind | Date | Download File |
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2250 | 130bit | Priebel | 584 | 575 | 6,120 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |

2251 | 145bit | Priebel | 1,002 | 993 | 11,315 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |

2252 | 162bit | Priebel | 3,606 | 3,597 | 37,118 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |

2253 | 176bit | Priebel | 7,441 | 7,431 | 82,270 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |

2254 | 192bit | Priebel | 13,691 | 13,682 | 154,303 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |

2255 | 208bit | Priebel | 24,430 | 24,421 | 299,756 | Combinatorial Problem | 2009 | MATLAB Rutherford Boeing Matrix Market |