The Multiple Polynomial Quadratic Sieve

Abstract

Abstract. A modification, due to Peter Montgomery, of Pomerance's Quadratic Sieve for factoring large integers is discussed along with its implementation. Using it, allows factorization with over an order of magnitude less sieving than the basic algorithm. It enables one to factor numbers in the 60-digit range in about a day, using a large minicomputer. The algorithm has features which make it well adapted to parallel implementation.

Factoring: Algorithms, computations, and computers

“…A further variation has been supplied by Peter Montgomery and Robert Silverman [26]. Instead of using only the single polynomial x 2 -N to generate quadratic residues which are small, they shift rapidly through multiple polynomials chosen so as to guarantee that the integers produced are all quadratic residues and are small.…”

MIMD-Factorisation on hypercubes

“…We assume familiarity with the 'Multiple Polynomial Quadratic Sieve' (MPQS) algorithm [8,11] and will sketch only the improved hypercube variation used by Now for every prime p in the factorbase let tv be a square root of N modp: t 2 ~ N modp. If p does not divide a, then…”

“…Of course, a parallel version of the MPQS algorithm, based on ideas of Dixon, Pomerance and Montgomery and described in [8,11], seemed to be the right point to start. However, a straightforward approach with processors running independently as in [3] is impossible because of memory constraints.…”

Parallel implementation of the quadratic sieve

“…The version of the Quadratic Sieve used in this implementation was fully described in Silverman [1987]. It differs from the original algorithm by using multiple polynomials rather than just a single polynomial.…”