Quadratic sieve; factoring a 192bit number. D. Priebel, Tenn. Tech Univ
Name 192bit
Group Priebel
Matrix ID 2254
Num Rows 13,691
Num Cols 13,682
Nonzeros 154,303
Pattern Entries 154,303
Kind Combinatorial Problem
Symmetric No
Date 2009
Author D. Priebel
Editor T. Davis
Structural Rank 13,006
Structural Rank Full false
Num Dmperm Blocks 1,903
Strongly Connect Components 590
Num Explicit Zeros 0
Pattern Symmetry 0%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type binary
SVD Statistics
Matrix Norm 1.200496e+02
Minimum Singular Value 2.043076e-63
Condition Number 5.875923e+64
Rank 13,006
sprank(A)-rank(A) 0
Null Space Dimension 676
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 = 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)