General number field sieve

In <a href="/facts/Number_theory/zhoa7zrc">number theory</a>, the general number field sieve (GNFS) is the most <a href="/facts/Algorithmic_efficiency/VutjvPTd">efficient</a> classical <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> known for <a href="/facts/Integer_factorization/EjMCTz2t">factoring integers</a> larger than <a href="/facts/Googol/FPxsLCU8">10100</a>. <a href="/facts/Heuristic/BcBEkv9c">Heuristically</a>, its <a href="/facts/Computational_complexity_theory/etHuVF3U">complexity</a> for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form

exp
                ⁡
                
                  (
                  
                    
                      (
                      
                        (
                        64
                        
                          /
                        
                        9
                        
                          )
                          
                            1
                            
                              /
                            
                            3
                          
                        
                        +
                        o
                        (
                        1
                        )
                      
                      )
                    
                    
                      
                        (
                        
                          log
                          ⁡
                          n
                        
                        )
                      
                      
                        1
                        
                          /
                        
                        3
                      
                    
                    
                      
                        (
                        
                          log
                          ⁡
                          log
                          ⁡
                          n
                        
                        )
                      
                      
                        2
                        
                          /
                        
                        3
                      
                    
                  
                  )
                
              
            
            
              
                =

L
                  
                    n
                  
                
                
                  [
                  
                    1
                    
                      /
                    
                    3
                    ,
                    (
                    64
                    
                      /
                    
                    9
                    
                      )
                      
                        1
                        
                          /
                        
                        3
                      
                    
                  
                  ]
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&\exp \left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)\\[5pt]={}&L_{n}\left[1/3,(64/9)^{1/3}\right]\end{aligned}}}

in <a href="/facts/Big_O_notation/weFFjSWg">O</a> and <a href="/facts/L-notation/gW7cKdVd">L-notations</a>. It is a generalization of the <a href="/facts/Special_number_field_sieve/vnlM5Tb7">special number field sieve</a>: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from <a href="/facts/Prime_power/R0vo3uYp">prime powers</a> (which are trivial to factor by taking roots). 
The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler <a href="/facts/Rational_sieve/XqojKpr6">rational sieve</a> or <a href="/facts/Quadratic_sieve/xDrfU9xl">quadratic sieve</a>. When using such algorithms to factor a large number n, it is necessary to search for <a href="/facts/Smooth_number/pC6DVLJW">smooth numbers</a> (i.e. numbers with small prime factors) of order n1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to achieve this speed-up, the number field sieve has to perform computations and factorizations in <a href="/facts/Number_field/e0K0TXSh">number fields</a>. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve.
The size of the input to the algorithm is log2 n or the number of bits in the binary representation of n. Any element of the order nc for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input.

General number field sieve open-in-new

General number field sieve