On the coefficients of the polynomial in the number field sieve

Abstract

Polynomial selection is very important in the number field sieve. If the number of relations a pair of polynomials can generate is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first, which can speed up the selection of good polynomials. In this paper, we aim to study the correlation between polynomial coefficients and the number of relations the polynomials can generate. By analyzing the zero roots, it is found that a polynomial with the ending coefficient containing more small primes usually can generate more relations than the one whose ending coefficient contains less. As a polynomial with more real roots usually can generate more relations, using the complete discrimination system, the requirements on the coefficients of a polynomial to obtain more real roots are analyzed. For instance, a necessary condition for a polynomial of degree d to have d distinct real roots is that the coefficient of degree d−2 should be negative or small enough. The result in the case d = 3 can be used directly in selecting polynomials generated by the nonlinear method, where d = 3 is already enough for practical purpose.

摘要

创新点

• 1. 提出新的研究角度, 研究多项式系数和其产出之间的关系。

• 2. 通过分析零根, 我们发现尾系数和首系数一样, 当其含有较多小因子时该多项式往往可以产生更多的关系。对于一个给定的大数, 是应该增大首系数, 还是尾系数? 这导致后续其他研究。

• 3. 利用CDS, 研究多项式有多个实根时其系数特点。比如, d次多项式有d个不同实根的一个必要条件是其d-2次系数要足够小。当d=3时, 该结论可用于非线性方法多项式的筛选, 而3次对于非线性方法已经足够了。