A Description of the Number Field Sieve

The number field sieve is a relatively new method to factor large integers. Its most notable success is the factorization of the ninth Fermat number. It is significantly faster than all known existing integer factoring algorithms. We examine the theoretical underpinnings of the sieve; after understanding how it works, we state the algorithm. We look mostly to the algebraic number theory aspects of the sieve while leaving the question of the sieve’s computational efficiency to other discussions.