Improving Polynomial Choice
The choice of polynomial can dramatically affect the time to complete the remainder of the algorithm. The method of choosing polynomials based on the expansion of n in base m shown above is suboptimal in many practical situations, leading to the development of better methods.
One such method was suggested by Murphy and Brent; they introduce a two-part score for polynomials, based on the presence of roots modulo small primes and on the average value that the polynomial takes over the sieving area.
The best reported results were achieved by the method of Thorsten Kleinjung, which allows g(x) = ax + b, and searches over a composed of small prime factors congruent to 1 modulo 2d and over leading coefficients of f which are divisible by 60.
Read more about this topic: General Number Field Sieve
Famous quotes containing the words improving and/or choice:
“She was so overcome by the splendor of his achievement that she took him into the closet and selected a choice apple and delivered it to him, along with an improving lecture upon the added value and flavor a treat took to itself when it came without sin through virtuous effort. And while she closed with a Scriptural flourish, he “hooked” a doughnut.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The question of place and climate is most closely related to the question of nutrition. Nobody is free to live everywhere; and whoever has to solve great problems that challenge all his strength actually has a very restricted choice in this matter. The influence of climate on our metabolism, its retardation, its acceleration, goes so far that a mistaken choice of place and climate can not only estrange a man from his task but can actually keep it from him: he never gets to see it.”
—Friedrich Nietzsche (1844–1900)