The number field sieve

The number field sieve is an algorithm to factor integers of the form $r^e-s$ for small positive $r$ and $s$. The authors present a report on work in progress on this algorithm. They informally describe the algorithm, discuss several implementation related aspects, and present some of the factorizations obtained so far. They also mention some solutions to the problems encountered when generalizing the algorithm to general integers using an idea of Buhler and Pomerance. It is not unlikely that this leads to a general purpose factoring algorithm that is asymptotically substantially faster than the fastest factoring algorithms known so far, like the multiple polynomial quadratic sieve