Description: Quadratic sieve; factoring a 162bit number. D. Priebel, Tenn. Tech Univ
|(bipartite graph drawing)|
|number of rows||3,606|
|number of columns||3,597|
|structural full rank?||no|
|# of blocks from dmperm||376|
|# strongly connected comp.||122|
|explicit zero entries||0|
|nonzero pattern symmetry||0%|
|numeric value symmetry||0%|
|Additional fields||size and type|
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 = 3408489886335277144344023699527218196631767672957 (162-bits) passes primality test, n is composite, continuing... 1) Initial bound: 80000, pi(80000) estimate: 7086, largest found: 71549 (actual bound) 2) Number of quadratic residues estimate: 4725, actual number found: 3596 3) Modular square roots found: 7192(2x residues) 4) Constructing smooth number list [sieving] (can take a while)... Sieving for: 3606 5. Constructing a matrix of size: 3606x3597 Set a total of 37118 exponents, with 1815 negatives Matrix solution found with: 1474 combinations Divisor: 1816046478796474796528999 (probably prime) Divisor: 1876873706775468629074043 (probably prime)
|nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD||1,707,729|
|nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD||1,647,739|
|null space dimension||137|
|full numerical rank?||no|
|singular value gap||8.11879e+12|
|singular values (MAT file):||click here|
|SVD method used:||s = svd (full (A)) ;|
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.