On Nonlinear Polynomial Selection and Geometric Progression (mod N) for Number Field Sieve.
Namhun Koo, Gooc Hwa Jo, Soonhak Kwon · IACR Cryptology ePrint Archive · 2011
The general number field sieve (GNFS) is asymptotically the fastest known factoring algorithm. One of the most important steps of GNFS is to select a good polynomial pair. A standard way of polynomial selection (being used in factoring RSA challenge numbers) is to select a nonlinear polynomial for algebraic sieving and a linear polynomial for rational sieving. There is another method called a nonlinear method which selects two polynomials of the same degree greater than one. In this paper, we generalize Montgomery’s method [7] using small geometric progression (GP) (mod N) to construct a pair of nonlinear polynomials. We introduce GP of length d + k with 1 ≤ k ≤ d − 1 and show that we can construct polynomials of degree d having common root (mod N), where the number of such polynomials and the size of the coefficients can be precisely determined.