Quadratic sieve; factoring a 208bit number. D. Priebel, Tenn. Tech Univ
Name 208bit
Group Priebel
Matrix ID 2255
Num Rows 24,430
Num Cols 24,421
Nonzeros 299,756
Pattern Entries 299,756
Kind Combinatorial Problem
Symmetric No
Date 2009
Author D. Priebel
Editor T. Davis
Structural Rank 22,981
Structural Rank Full false
Num Dmperm Blocks 3,671
Strongly Connect Components 1,231
Num Explicit Zeros 0
Pattern Symmetry 0%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type binary
SVD Statistics
Matrix Norm 1.908958e+02
Minimum Singular Value 0
Condition Number Inf
Rank 22,981
sprank(A)-rank(A) 0
Null Space Dimension 1,440
Full Numerical Rank? no
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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 = 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)