On Quadratic Polynomials for the Number Field Sieve

The newest, and asymptotically the fastest known integer factorisation algorithm is the number eld sieve. The area in which the number eld sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method nds quadratic polynomials. In this paper we examine the smoothness properties of integer values taken by these polynomials. Given a quadratic NFS polynomial f, let be its discriminant. We show that a prime p can divide values taken by f only if (=p) = 1. We measure the eeect of this residuosity property on the smoothness of f-values by adapting a parameter , developed for analysis of MPQS, to quadratic NFS poly-nomials. We estimate the yield of smooth values for these polynomials as a function of , and conclude that practical changes in might bring signiicant changes in the yield of smooth and almost smooth polynomial values.