Matrix: Priebel/176bit

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

 (bipartite graph drawing)

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 Matrix properties number of rows 7,441 number of columns 7,431 nonzeros 82,270 structural full rank? no structural rank 7,110 # of blocks from dmperm 893 # strongly connected comp. 282 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 7432-by-1 smooth_number full 7441-by-27 solution full 2983-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 = 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)
```

 Ordering statistics: result nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 6,717,048 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 6,250,729

 SVD-based statistics: norm(A) 96.1924 min(svd(A)) 1.31948e-47 cond(A) 7.29019e+48 rank(A) 7,110 sprank(A)-rank(A) 0 null space dimension 321 full numerical rank? no singular value gap 1.95793e+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.