Matrix: Priebel/192bit

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

Priebel/192bit graph
(bipartite graph drawing)

Priebel/192bit dmperm of Priebel/192bit
scc of Priebel/192bit

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  • Matrix group: Priebel
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  • download as a MATLAB mat-file, file size: 424 KB. Use UFget(2254) or UFget('Priebel/192bit') in MATLAB.
  • download in Matrix Market format, file size: 607 KB.
  • download in Rutherford/Boeing format, file size: 525 KB.

    Matrix properties
    number of rows13,691
    number of columns13,682
    structural full rank?no
    structural rank13,006
    # of blocks from dmperm1,903
    # strongly connected comp.590
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorD. Priebel
    editorT. Davis
    kindcombinatorial problem
    2D/3D problem?no

    Additional fieldssize and type
    factor_basefull 13682-by-1
    smooth_numberfull 13691-by-29
    solutionfull 5210-by-1


    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                                                 
    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 COLAMD20,360,732
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD18,023,441

    SVD-based statistics:
    null space dimension676
    full numerical rank?no
    singular value gap1.3401e+11

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

    Priebel/192bit svd

    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.