Matrix: Priebel/192bit

Description: Quadratic sieve; factoring a 192bit number. D. Priebel, Tenn. Tech Univ

 (bipartite graph drawing)

• Matrix group: Priebel
• download as a MATLAB mat-file, file size: 424 KB. Use UFget(2254) or UFget('Priebel/192bit') in MATLAB.

 Matrix properties number of rows 13,691 number of columns 13,682 nonzeros 154,303 structural full rank? no structural rank 13,006 # of blocks from dmperm 1,903 # strongly connected comp. 590 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 13682-by-1 smooth_number full 13691-by-29 solution full 5210-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 = 4232562527578032866150921497850842593296760823796443077101 (192-bits)
passes primality test, n is composite, continuing...
1) Initial bound: 350000, pi(350000) estimate: 27417,
largest found: 317729 (actual bound)
2) Number of quadratic residues estimate: 18279, actual number found: 13681
3) Modular square roots found: 27362(2x residues)
4) Constructing smooth number list [sieving] (can take a while)...
Sieving for: 13691
5. Constructing a matrix of size: 13691x13682
Set a total of 154303 exponents, with 6893 negatives
Matrix solution found with: 5210 combinations
Divisor: 83135929635332984850508004533 (probably prime)
Divisor: 50911351399373573113182167897 (probably prime)

 Ordering statistics: result nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD 20,360,732 nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD 18,023,441

 SVD-based statistics: norm(A) 120.05 min(svd(A)) 2.04308e-63 cond(A) 5.87592e+64 rank(A) 13,006 sprank(A)-rank(A) 0 null space dimension 676 full numerical rank? no singular value gap 1.3401e+11

 singular values (MAT file): click here SVD method used: s = svd (full (A)) ; status: ok