Matrix: Priebel/208bit

Description: Quadratic sieve; factoring a 208bit number. D. Priebel, Tenn. Tech Univ

Priebel/208bit graph
(bipartite graph drawing)


Priebel/208bit dmperm of Priebel/208bit
scc of Priebel/208bit

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  • download as a MATLAB mat-file, file size: 815 KB. Use UFget(2255) or UFget('Priebel/208bit') in MATLAB.
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    Matrix properties
    number of rows24,430
    number of columns24,421
    nonzeros299,756
    structural full rank?no
    structural rank22,981
    # of blocks from dmperm3,671
    # strongly connected comp.1,231
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    typebinary
    structurerectangular
    Cholesky candidate?no
    positive definite?no

    authorD. Priebel
    editorT. Davis
    date2009
    kindcombinatorial problem
    2D/3D problem?no

    Additional fieldssize and type
    factor_basefull 24421-by-1
    smooth_numberfull 24430-by-32
    solutionfull 8741-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 = 239380926372595066574100671394554319947805305453767699448870971 (208-bits)
    passes primality test, n is composite, continuing...                          
    1) Initial bound: 650000, pi(650000) estimate: 48562,                         
        largest found: 592903 (actual bound)                                      
    2) Number of quadratic residues estimate: 32376, actual number found: 24420   
    3) Modular square roots found: 48840(2x residues)                             
    4) Constructing smooth number list [sieving] (can take a while)...            
    Sieving for: 24430                                                            
    5. Constructing a matrix of size: 24430x24421                                 
    Set a total of 299756 exponents, with 12070 negatives                         
    Matrix solution found with: 8741 combinations                                 
    Divisor: 12216681953629826483019726942851 (probably prime)                    
    Divisor: 19594594283554225566102143686121 (probably prime)                    
    

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD63,577,458
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD53,620,282

    SVD-based statistics:
    norm(A)190.896
    min(svd(A))0
    cond(A)Inf
    rank(A)22,981
    sprank(A)-rank(A)0
    null space dimension1,440
    full numerical rank?no
    singular value gapInf

    singular values (MAT file):click here
    SVD method used:s = svd (full (R)) ; where [~,R,E] = spqr (A) with droptol of zero
    status:ok

    Priebel/208bit 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.