%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/Priebel/192bit
% name: Priebel/192bit
% [Quadratic sieve; factoring a 192bit number. D. Priebel, Tenn. Tech Univ]
% id: 2254
% date: 2009
% author: D. Priebel
% ed: T. Davis
% fields: name title A id date author ed kind notes aux
% aux: factor_base smooth_number solution
% kind: combinatorial problem
%-------------------------------------------------------------------------------
% 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 = 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)                    
%-------------------------------------------------------------------------------
