How to Factor More Effectively

Abstract

In Chap. 11 we described several factoring methods. Each will succeed in factoring some integers, but none of these is a state-of-the-art method that we would expect to succeed on a well-chosen RSA \(N = pq\). Even the best of these, CFRAC, suffers from the need to do trial division that will fail most of the time to provide any forward motion toward factoring N. In this chapter we discuss sieve methods for factoring. The primary computational benefit of a sieve method is that all the computational steps taken actually work toward finding factors, and that a sieve, stepping at constant stride through an array in memory, is highly efficient at the very lowest levels of a computing process. We discuss the Quadratic Sieve and the Multiple Polynomial Quadratic Sieve, and then finish with a nod to the current best method for factoring large “hard” integers, the Number Field Sieve.