Matrix: Priebel/162bit

Description: Quadratic sieve; factoring a 162bit number. D. Priebel, Tenn. Tech Univ

Priebel/162bit graph
(bipartite graph drawing)

Priebel/162bit dmperm of Priebel/162bit
scc of Priebel/162bit

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  • download as a MATLAB mat-file, file size: 103 KB. Use UFget(2252) or UFget('Priebel/162bit') in MATLAB.
  • download in Matrix Market format, file size: 143 KB.
  • download in Rutherford/Boeing format, file size: 117 KB.

    Matrix properties
    number of rows3,606
    number of columns3,597
    structural full rank?no
    structural rank3,460
    # of blocks from dmperm376
    # strongly connected comp.122
    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 3597-by-1
    smooth_numberfull 3606-by-25
    solutionfull 1474-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 = 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)                      

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD1,707,729
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD1,647,739

    SVD-based statistics:
    null space dimension137
    full numerical rank?no
    singular value gap8.11879e+12

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

    Priebel/162bit 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.