Experimenting with NFS Sample Clauses

Experimenting with NFS. We now compare the theoretical densities with the practical densities of the semi- smooth numbers in the case of the number field sieve. This is more complicated than the comparison for MPQS, as there are more aspects that we must take into account. We start with a similar approach as in Subsection 3.3.1, but we give as well the better approach (already mentioned in a footnote in Section 3.3) that takes more polynomial values into account (as in [29]) by dividing the sieving region into subregions.
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Experimenting with NFS. We now compare the theoretical densities with the practical densities of the semi- smooth numbers in the case of the number field sieve. This is more complicated than the comparison for MPQS, as there are more aspects that we must take into account. We start with a similar approach as in Subsection 3.3.1, but we give as well the better approach (already mentioned in a footnote in Section 3.3) that takes more polynomial values into account (as in [29]) by dividing the sieving region into subregions. | | In NFS we work with two homogeneous polynomials F1(x, y) and F2(x, y), and we search for pairs (a, b) with gcd(a, b) = 1 and b > 0 such that both |F1(a, b)| and F2(a, b) are (semi)smooth. For both polynomials we are looking for the highest polynomial value in the sieving region and if necessary we apply a correction for the average value and ▇▇▇ ▇▇▇▇▇ of Fi(a, b), for i = 1, 2, modulo primes in the factorbase (similar to the correction we applied for the polynomials in MPQS). We assume that the probability of F1(a, b) being semismooth is independent of the probability of F2(a, b) being semismooth. − First we give our results for a number (19, 183 , C131) we factored with the special number field sieve. The polynomials are • F1(x, y) = 37589973457545958193355601x − y, and
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Related to Experimenting with NFS

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