%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/Priebel/208bit
% name: Priebel/208bit
% [Quadratic sieve; factoring a 208bit number. D. Priebel, Tenn. Tech Univ]
% id: 2255
% 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 = 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)                    
%-------------------------------------------------------------------------------
