# ON NONLINEAR POLYNOMIAL SELECTION AND GEOMETRIC PROGRESSION (MOD N) FOR NUMBER FIELD SIEVE

• Cho, Gook Hwa (Institute for Mathematical Sciences Ewha Womans University) ;
• Koo, Namhun (Division of Mathematical Models National Institute for Mathematical Sciences) ;
• Kwon, Soonhak (Department of Mathematics Sungkyunkwan University)
• Published : 2016.01.31
• 92 8

#### Abstract

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 [12] using geometric progression (GP) (mod N) to construct a pair of nonlinear polynomials. We also introduce GP of length d + k with $1{\leq}k{\leq}d-1$ 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.

#### Keywords

polynomial selection;number field sieve;geometric progression;LLL algorithm

#### Acknowledgement

Grant : BK21플러스

Supported by : 성균관대학교

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