**Matrix: Priebel/145bit**

Description: Quadratic sieve; factoring a 145bit number. D. Priebel, Tenn. Tech Univ

(bipartite graph drawing) |

Matrix properties | |

number of rows | 1,002 |

number of columns | 993 |

nonzeros | 11,315 |

structural full rank? | no |

structural rank | 964 |

# of blocks from dmperm | 119 |

# strongly connected comp. | 28 |

explicit zero entries | 0 |

nonzero pattern symmetry | 0% |

numeric value symmetry | 0% |

type | binary |

structure | rectangular |

Cholesky candidate? | no |

positive definite? | no |

author | D. Priebel |

editor | T. Davis |

date | 2009 |

kind | combinatorial problem |

2D/3D problem? | no |

Additional fields | size and type |

factor_base | full 993-by-1 |

smooth_number | full 1002-by-22 |

solution | full 385-by-1 |

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 = 27393004579711727757848513391018843988362569 (145-bits) passes primality test, n is composite, continuing... 1) Initial bound: 20000, pi(20000) estimate: 2019, largest found: 17569 (actual bound) 2) Number of quadratic residues estimate: 1347, actual number found: 992 3) Modular square roots found: 1984(2x residues) 4) Constructing smooth number list [sieving] (can take a while)... Sieving for: 1002 5. Constructing a matrix of size: 1002x993 Set a total of 11315 exponents, with 503 negatives Matrix solution found with: 385 combinations Divisor: 4762476283061573160587 (probably prime) Divisor: 5751840629031342254587 (probably prime)

Ordering statistics: | result |

nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD | 139,428 |

nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD | 150,622 |

SVD-based statistics: | |

norm(A) | 39.2964 |

min(svd(A)) | 2.03061e-32 |

cond(A) | 1.9352e+33 |

rank(A) | 964 |

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

null space dimension | 29 |

full numerical rank? | no |

singular value gap | 4.50578e+13 |

singular values (MAT file): | click here |

SVD method used: | s = svd (full (A)) ; |

status: | ok |

For a description of the statistics displayed above, click here.

*Maintained by Tim Davis, last updated 12-Mar-2014.Matrix pictures by cspy, a MATLAB function in the CSparse package.
Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.
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