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On Quadratic Polynomials for the Number Field Sieve
 Australian Computer Science Communications
, 1997
"... . The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this pa ..."
Abstract

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