Quadratic sieve; factoring a 130bit number. D. Priebel, Tenn. Tech Univ
Name 130bit
Group Priebel
Matrix ID 2250
Num Rows 584
Num Cols 575
Nonzeros 6,120
Pattern Entries 6,120
Kind Combinatorial Problem
Symmetric No
Date 2009
Author D. Priebel
Editor T. Davis
Structural Rank 566
Structural Rank Full false
Num Dmperm Blocks 45
Strongly Connect Components 9
Num Explicit Zeros 0
Pattern Symmetry 0%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type binary
SVD Statistics
Matrix Norm 3.052082e+01
Minimum Singular Value 7.224610e-17
Condition Number 4.224563e+17
Rank 566
sprank(A)-rank(A) 0
Null Space Dimension 9
Full Numerical Rank? no
Download Singular Values MATLAB
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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 = 787911249838484617926390474950774839909 (130-bits)                   
passes primality test, n is composite, continuing...                     
1) Initial bound: 10000, pi(10000) estimate: 1085,                       
    largest found: 8707 (actual bound)                                   
2) Number of quadratic residues estimate: 725, actual number found: 574  
3) Modular square roots found: 1148(2x residues)                         
4) Constructing smooth number list [sieving] (can take a while)...       
Sieving for: 584                                                         
5. Constructing a matrix of size: 584x575                                
Matrix solution found with: 261 combinations                             
Divisor: 23850290715477455017 (probably prime)                           
Divisor: 33035708421246870877 (probably prime)